Despite all the movies and stories based around the idea of hiding an asteroid impact from the public, there is no possibility of that at all in real life. The night sky is open to anyone to observe if they have a telescope. There are many big telescopes managed by dozens of separate countries world wide of many different political persuasions. There are thousands of professional astronomers and probably millions of amateurs who are keen enough to have big telescopes of their own.
Also the way asteroid discovery is done is that as soon as anyone spots a new asteroid, they release these observations publicly to the minor planet center. All interested astronomers, including keen amateurs, can then point their telescope at the newly discovered asteroid. It takes big custom designed telescopes with huge mirrors and vast CCD chips to find the asteroids nowadays, because they have to scan large areas of the sky every night. Such telescopes cost many millions of dollars, and are way beyond the capability of amateur telescopes, which have to focus on just one spot in the sky. The Hubble space telescope is no good for this either, as its field of view is far too small. At present Pan STARRS is our best asteroid detection telescope though in 2022 then the large synoptic survey telescope will go into the lead when it starts work. These are telescopes managed by a consortium of many countries. There are several other large scale searches that find many asteroids, using ground based telescopes. NEOWise also helps, a space infrared telescope, so space telescopes can also help but only if they have a wide field of view and especially if they are able to detect infrared as asteroids are brightest in infrared.
So anyway - amateurs used to find many of the asteroids too, but nowadays just about all discoveries are made by these big wide field huge mirror telescopes with enormous CCD chips. But once you detect the asteroid, that’s only the beginning. If you have a single image, you have no way to know what direction it is moving, and Earth is a miniscule target, like trying to hit a single grain of dust in a large room by throwing another even tinier grain of dust, so small as to be invisible, in its direction. We have discovered over 16,000 asteroids to date that do flybys of Earth and potentially could hit us, including 876 larger than 1 km - that’s more than 90% of the ones larger than 1 km, and we are already tracking all the ones of over 10 km that have any chance of hitting Earth. We are tracking over 7,700 of 140 meters or larger. You can look up the latest figures here Discovery Statistics
None of those asteroids will hit Earth before 2200, which helps underscore how hard it is for a random asteroid to hit Earth. So detecting the asteroid is just the first step. It’s almost certain to miss us, every time it does a flyby, right through to 2200 and beyond. You then have to find its orbit. Theoretically, three observations are enough, if widely separated. But with Earth such a tiny target, in practice you need many more than that, to pin down the orbit enough to be able to tell if this even tinier asteroid could hit that tiny dust grain of a planet decades into the future. There everyone joins in, as once it is discovered, the amateur astronomers can track it too, for as long as it remains within range - as the discovery telescopes don’t use enormously high magnifications like the amateurs. The large telescopes do some follow up observation too - for instnace, next time that patch of sky comes around in the survey they will photograph the asteroid wherever it has got to, but their main objective is discovery, not follow up.
So the usual pipline is that the discovery telescopes survey large areas of the sky quickly, with very sensitive CCDs, then the amateurs, together with some professionals, look closely at each patch where an asteroid has been detected and publish their observations again through the minor planet center. Finally the geeky orbit calculators work out the orbit from the observations, predict where to look to observe it in the future, they also trace the orbit backwards so that you can look in pre-discovery images to see if anyone spotted it in the past and didn’t recognize it as an asteroid. As this goes on, you get more and more refined orbit calculations, better observations, and eventually you figure out if it can hit or not.
Sometimes at early stages, they figure out that it has a tiny chance of hitting Earth, and if that happens you get a “yellow alert” level, increased interest, everyone interested with a telescope would be working on it. Probably also they would schedule radar observations during close flybys too, using large radio telescopes like the Arecibo Observatory (if it passes within its field of view) . That’s especially useful if they can do it, because radar gives very precise distance measurements. Then as they refine the orbit, normally they find out it is going to miss Earth, as you’d expect, since the chance of hitting was tiny all along. If we get a yellow alert, as we surely will again in the future at some point, expect it to be downgraded to a miss as they find out more, as by far the most likely outcome.
All this is played out in public. If an asteroid has a tiny chance of hitting Earth, this becomes headline astronomy news. If it had a significant chance, I guarantee it, it would be world wide news on all the news networks. Experts would be discussing it, we’d be kept updated with all the latest theories and discoveries and updates on the latest probability estimates, and calls for amateurs to photograph it and send in their observations. If it was going to impact Earth and there was nothing we could do about it, they would say exactly where it would impact, and we’d get evacuations of the impact zone. The actual impact crater of a 1 km asteroid, which is about the largest that could catch us unawares with only months, or more likely years of warning, would be 10 km in diameter. So you’d definitely have at least a 10 km diameter region to evacuate, but probably much larger than that, if it was predicted to fall on the land.
There’d be large scale effects beyond that, blast damage, strong winds, broken windows, and there’d be some global effects like a large volcanic eruption. The climate might be cooler for a few years affecting crops. With some warning, you could prepare for that too, stock pile crops, work out mitigation levels etc.
That’s the worst that could happen, but it is very very unlikely. Anyway in that situation we would evacuate the impact area - where the crater forms. The crater s very unlikely to overlap a city, and would most probably be in a desert area or in the sea since most of the Earth’s surface is sea or desert. If in the sea, a 1 km asteroid impact is large enough for a tsunami, but a predictable tsunami - you’d know when and where it would happen, months in advance or at least weeks in advance, exactly to the minute. Everyone would know to keep away from the sea shore at that time, and they’d have predictions of how high it would be, so how far away you’d need to go, again months or weeks before it happened.
But this is very very unlikely. In the whole of human history we’ve never had an impact like that. We haven’t even had one of the 140 meter diameter impacts that would destroy a city if it hit one, and cause severe damage to a whole small country with the gusts of wind. And the risk this century is if anything less than in previous centuries, because we are tracking over 7,000 of those 140 meter asteroids and know none of those are headed our way. There are many more to discover, but the more we discover, the less our risk is, for the next couple of centuries, unless of course we find one that is going to hit, in which case it becomes a known hazard we can prepare for and work to deflect if possible.
Also, if we find a large asteroid headed our way, it’s far more likely to just miss Earth and do a close flyby than to hit, first time it flies past Earth, and to have a predicted hit decades into the future. That’s just because Earth is so tiny again, so a flyby is far more likely than a hit, so it does many flybys before it hits, so we are more likely to catch a potential impactor just before a flyby than just before a hit. In that case, we can then deflect it, with many ways to do it given decades of warning. You need only centimeters per second change in its speed, or even less if it does a flyby first, the most likely case.
The most likely size of impact to predict though is 20 meters upwards, maybe 40 meters in diameter. That’s because the numbers of asteroids increase hugely as the size gets smaller, so in fact, nearly all asteroids that fly past Earth are of this tiny category of 20 meters upwards. By comparison, the larger ones are rare. This is similar in size to Chelyabinsk or a bit larger.
They also are rare. Perhaps once every 80 years - and again most often will happen in the sea or over an uninhabited area of the Earth. These are too small for a tsunami, so we don’t need to worry about impacts in the sea, except to keep shipping and planes away from the impact zone during the minute of impact and a short while afterwards. Anyway - the very small 20 meter ones could slip through the net still and hit us undetected even up to the last minute like Chelyabinsk, if the final approach is from the direction of the Sun. Even the Chelyabinsk one, if it had been just a bit larger, say 40 meters, would have been discovered months in advance.
So - the most likely future headline is a predicted impact of a 40 meter diameter asteroid, with a lead time most likely of at least months, or it could be years or decades. It’s most likely to hit the sea. If it hits land, it’s most likely to be predicted to hit an uninhabited desert, or a sparsely inhabited area, so that at most, a few nomads or villagers might need to be evacuated. That’s the most likely scenario for the next predicted impact with consequences for humans. At a much lower level of probability, we have the chance that it hits a more populated area or a city.
At that size, it could definitely be deflected, given a lead time of years, especially if it does a previous flyby first. If discovered only weeks or months before impact, then it would be a case of evacuating the impact zone - it would only cause a small crater, but in worst case it could also cause an air burst just above ground level. So you might have to evacuate a region up to many square kilometers, as for the Tunguska blast. Beyond that region, then you’d warn people to keep away from windows during the event itself, and to beware of flying glass and protect themselves from it.
That’s the most likely future scenario. In any case the whole thing would play out in the public eye and there simply is no way it could be hidden, even if anyone wanted to. Basically because the night sky is open to anyone to observe and nobody can do anything about that, even if they wanted to. And they haven’t even tried. Indeed we’ve had several “dry runs” for this. Apophis briefly went right up to the topmost of the yellow alerts, and this was widely publicized at the time, and you could follow the successive announcements as they first had a slight increase in the alert level, then they found that it was going to miss Earth, but for a while had a chance of an impact at a second flyby - but all the time these predictions were low probability so it was no surprise to anyone that eventually they proved that it would miss.
During all of that, the new observations and calculations were simply published publicly as soon as they were known.
We’ve also had a couple of very small asteroids that hit Earth, in a remote desert in Africa, and in the sea off India, which were predicted to hit just hours before impact. Again that was all done in public.
So my answer is, first, yes of course it should be made public, because that way you can save many lives, and we can also all work together to find a way to deflect it, if that is at all possible. But there is no choice anyway. It all would play out in the public arena, no matter what choices anyone might try to make behind the scenes, and astronomers and governments world wide recognize this, so they wouldn’t even try to hide it, knowing that it is impossible to do that.
You can check if there is any predicted impact easily, go here, Sentry: Earth Impact Monitoring
Click on "Unconstrained settings", then, as it is sorted with the highest risk first, if the top entry is white, blue or green there is no predicted impact, or it is too small (if blue) to be of any consequence. If yellow, it's like a yellow alert that will most likely turn out to be a miss. If it is red, you'll know all about it anyway as it will be top of the news not just in astronomy news sites but on the mainstream media news too.