Many ETs are non technological. Like parrots or elephants, or dogs just not physically easily capable of making complex things however much they want to. Or live in sub surface oceans like Europa - if they live in the sea that also makes them likely to be non technological e.g. can't use fire easily. And may not know that the rest of the universe exists. Such an ET could even live in our solar system. We could have ETIs in the subsurface oceans of Europa, Ganymede, Titan and Encladus and if they were non technological we'd have no idea at all that they exist, and they wouldn't know that we exist. Given how many more icy moons there are than habitable planets in our solar system - these may well be the most common types of ETI in the galaxy.
The ones that are technological are either short sighted in which case they destroy themselves early on or at least their technology and don't remain technological for long. Or if long sighted - then they think carefully before colonizing the galaxy. Reason is - that if you colonize the galaxy - what are your colonies going to turn into? What creatures or machines will your descendants ten or a hundred generations down the line unleash onto the galaxy?
Especially bearing in mind that amongst billions of colonies separated by light speed barriers - some may "de-evolve" biologically or socially. May become beings of only baby like levels of understanding for instance, but with machines giving them enormous powers. Others may develop obscure ideas and aims that are baffling to everyone else, strange social or antisocial objectives, or may make huge technological or biological blunders. Which with galaxy shaping power, e.g. to unleash self replicating technology could literally trash the galaxy.
I think that a cautious ETI would give that considcrable thought and probably wouldn't start to colonize the galaxy until they had worked it through to their satisfaction.
By the time they have the capability to colonize - and also have developed to the point where socially they are able to colonize a galaxy without trashing it - confident that it will be in safe hands in their descendants even ten or a hundred or even a million generations away (and with long lives they may still be alive to see their hundred times removed grandchildren) - then they may have very different ideas about it than we do.
Would they even want to colonize? If they do, maybe their technology is so advanced that we can hardly recognize it as technology at all? Maybe what they use is so unlike what we currently call technology that it is more like magic for us? Probably they wouldn't swarm through the galaxy and fill every single planet and star system with their species, as humans surely would at our present stage of social development. Maybe they feel no hurry to fill the galaxy, if at all, and maybe the nearest colony is a thousand light years away (still very close on galactic scale). And maybe many ETIs simply don't want to colonize at all. But are rather - a bit like the ETs in ET the movie - they explore, visit places, sometimes take samples back with them - but leave things as they are. Our galaxy could be filled with millions of explorers like that, and if they aren't bent on colonizing it and filling it with their species, the chance that any will encounter Earth in the near future is probably tiny. Maybe our last visit, if we had one at all, was a hundred million years ago, and maybe due another visit some time in the next hundred million years. And if they do visit, then we'd only notice them if they wanted us to.
And - I totally don't believe at all in crashing ETI spaceships. Their technology has to be millions of years old, otherwise astonishing coincidence that they arise just at the same time as us. Their spacecraft wouldn't crash. And for that matter, they would be very unlikely to visit us in person. Instead send avatars, even if they are in orbit around Earth, why land on surface in person? And expose themselves to our biology and us to their biology - surely quite dangerous? And all the other hazards of an unknown planet? When they can experience it as well as, or even better, via telepresence.
WHY I THINK WE WILL SOON HAVE THE TECHNOLOGICAL CAPABILITY TO COLONIZE THE GALAXY
I'm not saying here that we will do it. But that we will have the capability to do it, probably.
However, we could also do it via slow colonization of the galaxy via the Oort clouds - an idea that dates back to Freeman Dyson and Carl Sagan. Colonization of trans-Neptunian objects. The idea there is that we will soon have fusion power, if we continue as a technological species. Maybe in the next few decades, but surely at some point in the next few centuries or millennia we will have this.
So once you have fusion power you have virtually unlimited power which you can get from ice or water or similar materials. So then you are no longer tied down to living close to a star, because you don't need sunlight for energy any more. You could just as easily live in the outer solar system or the Oort cloud. And then - the thing is that the Oort cloud stretches out all the way we think to the point where it mingles with the Oort clouds of nearby stars.
We are- nowhere near to being able to do this now but surely in next few centuries, or say a thousand years we will have that capability. As well as that, surely we will have self replicating machines, "clanking replicators" at least. Got most of the technology for that already, 3D printers, even nanoscale printers - once we have that fully developed then it would be almost effortless to build even large habitats and do technological mega projects which again would make colonizing the Oort cloud almost easy.
And then - once that genie is out of the bottle - there is nothing really to stop colonization of the entire galaxy, seems to me.
What's more, it probably would be quite a fast process too. Normally you think of colonization as averaging, maybe a hundredth or a thousandth of the speed of light. because you have to repeatedly stop and build a new civilization before you get to the point where you send out new spaceships again. At a thousandth of the speed of light you could colonize the entire galaxy, diameter 100,000 light years, in a hundred million years.
But if humans become independent of stars, and able to live almost anywhere between the stars in the Oort Cloud object stepping stones, it could be far faster than this.
You would soon have trillions of colonists, then quadrillions of them. And the expanding frontier would have spaceships leaving it continually at whatever is the fastest easily feasible speed - perhaps a tenth of the speed of light. Even if nearly everyone in the civilization travels at a far more sedate speed, you just need one in a billion of the colonists to do fast travel colonization and that's what would happen.
And what's more, they don't need to stop and rebuild when they reach the next object. The next Oort cloud object is only a few AUs away probably. Just stop, drop off a few colonists, load up with fuel and provisions, do any repairs, build a new spaceship from time to time, and keep going to the next one, quite possibly as soon as a few months later.
And with fusion power, practically unlimited energy for their rockets - they could easily just accelerate at full g, and so get to a tenth of the speed of light pretty quickly, after about a month of acceleration at 1 g. Striving for the Speed of Light. Or even sooner if you do hyper g.
With technology like that, they then could colonize the galaxy in a million years. Just a blink. An ETI could have colonized the entire galaxy several thousand times over, since the birth of our sun.
So - I think myself that at some point in the next thousand years, maybe sooner, we will have the technological capability to make a start on an irreversible colonization of the galaxy. But will we have the wisdom to do it? Will we know how to do it in a way that is responsible to the galaxy? We will have to decide at some point if we should do this.
If we don't want to colonize the galaxy right away, we could use the Oort clouds instead as a source of materials, especially ice for water - to build our habitats closer to the sun. That would help to keep us around this sun in a natural way. And have only explorers in the rest of the galaxy, not colonies, until we feel we have evolved enough to be able to colonize it responsibly. It would be easy enough, at that stage of civilization, to move to another star in any emergency, and for that matter, not much that could happen to us in our solar system once independent of planetary surfaces. So would be no great urgency to colonize for survival.
So - might be that that is what ETIs do if they are wise and responsible. And if not, they never get that far.
WHY I THINK HUMANS AND OTHER ETIS HAVE A GOOD CHANCE OF SURVIVING TECHNOLOGY
I think it depends on how reckless they are as a species. E.g. - think how different an ETI would be in character if it was biologically a parrot, or a sheep, or a dog, or elephant, or an octopus, or a dolphin or a goat (say) , or a cat. All quite intelligent animals but very different in general personality and character from each other as species (as well as individuals within species of course).
As for humans, basically we are great apes in temperament, a bit like gorillas. I'm optimistic that we may be far seeing enough and able to deal with our aggression enough to survive our technology too.
I think we've already passed through several hurdles that probably would have sent some other ETIs back to the stone age, such as biological weapons, nuclear weapons, and the ability to put weapons and military bases into outer space, e.g. nuclear weapons and military bases and orbiting missiles in LEO. We've got treaties to prevent us doing this - which actually work what's more.
Also have had many other problems which we solved by co-operation and technology. E.g. the green revolution for one, or dealing with the threats for the ozone layer ,saving whales, dealing with the threat to wild birds from DDT, setting aside Antarctica and various other parts of the world as scientific and nature preserves. Also success in eradicating smallpox, much progress in dealing with diseases, have done a lot to reduce poverty although not yet eradicated it, etc., etc.
We tend to look at our failures - and we have had our share of miserable failures - but we've also had many astonishing successes too. I think we have a pretty decent chance actually, overall.
SO WHAT IS OUR CHANCE OF SPOTTING THEM?
I think there's a chance that we could spot their technology if it is megatechnology type, or they use high energies, or spot them if they want to be seen and send out a beacon to the galaxy. If they don't make a special effort to be seen, and if they have efficient technology that doesn't leak big signals into space, even if technological - then they could be as close as the nearest stars and we probably wouldn't know they are there, if their technology is even a few thousand years ahead of us, and it would probably be millions of years ahead of us.
This is my own take on this and my own favourite solution. There are many other good answers to this question presenting many other solutions to the paradox so for a good idea of the range of ideas take a look at some of those.