Yes, astronomers think that probably most stars have Oort clouds around them, which extend much of the way to the nearest stars. And they probably also share objects with each other. Some of our own Oort cloud objects may originate around other stars. When stars first form in star clusters, they may be exchanging comets with each other, according to one hypothesis.
The Sun Steals Comets from Other Stars
But we don't have direct evidence yet of our own Oort cloud. Our telescopes are still just not quite sensitive to pick it up. We can only pick up objects that are diverted into the inner solar system, and hypothesize that they originated in the Oort cloud. And none of our spacecraft have reached it yet, not the Oort cloud proper:
.
None of our spacecraft will be functioning any more when they reach it, either, not the ones we've sent so far.
Incidentally though it used to be thought of as consisting entirely of icy objects, it may have many rocky asteroids as well. There may be more rocky asteroids in the Oort cloud than there are in the asteroid belt.
Eight Billion Asteroids in the Oort Cloud? - Sky & Telescope
There may be at least two kinds of object out there - the icy ones that may be reddish in colour originally like Pluto, and bluish ones that are like asteroids.
First observations of the surfaces of objects from the Oort Cloud
Nix (moon) - the reddish spot is similar in colour to Pluto. Some of the Oort cloud objects may be this colour.
So far I don't think there are any observations of Oort clouds around other stars. There are observations of exocomets - they were discovered before the first exoplanets, from observations of Beta Pictoris. Hundreds of Comets Seen Orbiting Distant Solar System.
And later were found around other stars as well. Hasn't been nearly as much work on finding exocomets as exoplanets but they may be very common alsoo 'Exocomets' Common Across Milky Way Galaxy
The problem is that the Oort cloud is so far away, Hubble can't even resolve planets out there. It starts at 1,000 AU. Why Can't we see the Oort Cloud?
The WISE survey looking for planets in the Oort cloud ruled out the possibility of ruled out the possibility of a Saturn sized object out to 10,000 times the Earth - Sun distance, and a Jupiter size or larger object out to 26,000 times that distance. X marks the... They SAID there was a mystery planet there – NASA.
And there is a hypothesis that there might be an Earth or Mars sized object a bit over 50 au from Earth and we can't yet prove or disprove it. So we simply can't see anything out there unless it is truly massive.
90377 Sedna has been hypothesized to originate in the inner Oort cloud, but there are many other ideas for its origin. It's now in a very eccentric orbit and the longest period of any of the large known objects in the solar system of about 11,400 years.
In principle, we could get indirect evidence of Oort clouds around other stars from studying their exocomets just as we get indirect evidence for our sun by studying our comets. We might also just possibly be able to observe the dust in the Oort cloud around another star. Page on caltech.edu
Sometimes you get comets pass through our own solar system that have such high velocity they might have come from another star. Jupiter sometimes ejects comets from the solar system, so other star systems presumably do the same.
The first observed comet that might have been ejected from our solar system was Lexell's Comet and this is back in the late eighteenth century. The great French mathematician Laplace showed that the reason it never reappeared in 1782 is that a series of encounters with Jupiter had changed its orbit and possibly ejected it from the solar system Villain in disguise: Jupiter’s role in impacts on Earth - Astrobiology Magazine
And the other way around, Jupiter could capture an Interstellar comet
But the problem there is that the fastest comets will be able to fly straight past and not be deflected much by our sun. The ones that are slow enough to be pulled towards our sun and do gravitational flybys would be so slow that they are rather hard to distinguish from ordinary comets in the very distant Oort cloud.
One prediction in 1991 was that 1 in 2,400 long period comets is truly exosolar hyperbolic. 1991JBAA..101..119H Page 120. But that was based on many assumptions.
Working back the other way, you can use the non detection of interstellar comets on their first flyby of the inner solar system to get an upper bound on the density of comets in interstellar space between the stars Page on iop.org
To confuse the matter, then you also have comets that appear to be hyperbolic but were originally elliptical and turned into hyperbolic orbits on their way into the inner solar system. comet | astronomy. When you just take account of gravitational effects, their original orbit is hyperbolic. But CO2 sublimation and H20 sublimation can turn an originally elliptical orbit into a hyperbolic one as it approaches the sun Page on aanda.org
There are many objects in hyperbolic (eccentricity greater than 1) or parabolic orbits. List of hyperbolic comets But I don't think we yet have a confirmed comet that definitely comes from another star. If we ever spot one with a high eccentricity such as say 2.0, or ones that we can prove had hyperbolic orbits originally before they came into the inner solar system it would be pretty good evidence for comets from other stars and we could start to get indirect evidence for Oort clouds around other stars that way.
By the way - there's an idea to use an exo comet ejected from our solar system for the first stages of an interstellar journey Exocomet Express: How to Hitch Intersellar Rides on Comets. Just came across that while searching for material for this answer, not sure how practical it is.
See also:
In short, very likely that many stars have Oort clouds. But so far not confirmed for sure that any star does, even our own, though we don't have any good alternative hypothesis to the Oort cloud to explain the distribution of comet orbits AFAIK.
But we don't have direct evidence yet of our own Oort cloud. Our telescopes are still just not quite sensitive to pick it up. We can only pick up objects that are diverted into the inner solar system, and hypothesize that they originated in the Oort cloud. And none of our spacecraft have reached it yet, not the Oort cloud proper:
.
Note this is a logarithmic plot - distances close to the sun are exaggerated. The outer edge of the hypothesized Oort cloud is 100,000 aus. Distance to Alpha Centauri at 4.367 light years is 276,000 aus. So (if it exists and is as hypothesized) the outer edge of the Oort cloud is a little over a third of the way to Alpha Centauri.
None of our spacecraft will be functioning any more when they reach it, either, not the ones we've sent so far.
Incidentally though it used to be thought of as consisting entirely of icy objects, it may have many rocky asteroids as well. There may be more rocky asteroids in the Oort cloud than there are in the asteroid belt.
Eight Billion Asteroids in the Oort Cloud? - Sky & Telescope
There may be at least two kinds of object out there - the icy ones that may be reddish in colour originally like Pluto, and bluish ones that are like asteroids.
First observations of the surfaces of objects from the Oort Cloud
So far I don't think there are any observations of Oort clouds around other stars. There are observations of exocomets - they were discovered before the first exoplanets, from observations of Beta Pictoris. Hundreds of Comets Seen Orbiting Distant Solar System.
Artist's impression of the Beta Pictoris Exocomet - and planet formation processes there - it may give a glimpse into processes that occurred during formation of our own solar system - and it may have an Oort cloud.
And later were found around other stars as well. Hasn't been nearly as much work on finding exocomets as exoplanets but they may be very common alsoo 'Exocomets' Common Across Milky Way Galaxy
The problem is that the Oort cloud is so far away, Hubble can't even resolve planets out there. It starts at 1,000 AU. Why Can't we see the Oort Cloud?
The WISE survey looking for planets in the Oort cloud ruled out the possibility of ruled out the possibility of a Saturn sized object out to 10,000 times the Earth - Sun distance, and a Jupiter size or larger object out to 26,000 times that distance. X marks the... They SAID there was a mystery planet there – NASA.
And there is a hypothesis that there might be an Earth or Mars sized object a bit over 50 au from Earth and we can't yet prove or disprove it. So we simply can't see anything out there unless it is truly massive.
90377 Sedna has been hypothesized to originate in the inner Oort cloud, but there are many other ideas for its origin. It's now in a very eccentric orbit and the longest period of any of the large known objects in the solar system of about 11,400 years.
In principle, we could get indirect evidence of Oort clouds around other stars from studying their exocomets just as we get indirect evidence for our sun by studying our comets. We might also just possibly be able to observe the dust in the Oort cloud around another star. Page on caltech.edu
Sometimes you get comets pass through our own solar system that have such high velocity they might have come from another star. Jupiter sometimes ejects comets from the solar system, so other star systems presumably do the same.
The first observed comet that might have been ejected from our solar system was Lexell's Comet and this is back in the late eighteenth century. The great French mathematician Laplace showed that the reason it never reappeared in 1782 is that a series of encounters with Jupiter had changed its orbit and possibly ejected it from the solar system Villain in disguise: Jupiter’s role in impacts on Earth - Astrobiology Magazine
And the other way around, Jupiter could capture an Interstellar comet
Comet Hyakutake C/1996 B2 which may possibly be an interstellar comet captured by Jupiter, as may be Machholz 1 . A Very Oddball Comet - Sky & Telescope
But the problem there is that the fastest comets will be able to fly straight past and not be deflected much by our sun. The ones that are slow enough to be pulled towards our sun and do gravitational flybys would be so slow that they are rather hard to distinguish from ordinary comets in the very distant Oort cloud.
One prediction in 1991 was that 1 in 2,400 long period comets is truly exosolar hyperbolic. 1991JBAA..101..119H Page 120. But that was based on many assumptions.
Working back the other way, you can use the non detection of interstellar comets on their first flyby of the inner solar system to get an upper bound on the density of comets in interstellar space between the stars Page on iop.org
To confuse the matter, then you also have comets that appear to be hyperbolic but were originally elliptical and turned into hyperbolic orbits on their way into the inner solar system. comet | astronomy. When you just take account of gravitational effects, their original orbit is hyperbolic. But CO2 sublimation and H20 sublimation can turn an originally elliptical orbit into a hyperbolic one as it approaches the sun Page on aanda.org
There are many objects in hyperbolic (eccentricity greater than 1) or parabolic orbits. List of hyperbolic comets But I don't think we yet have a confirmed comet that definitely comes from another star. If we ever spot one with a high eccentricity such as say 2.0, or ones that we can prove had hyperbolic orbits originally before they came into the inner solar system it would be pretty good evidence for comets from other stars and we could start to get indirect evidence for Oort clouds around other stars that way.
By the way - there's an idea to use an exo comet ejected from our solar system for the first stages of an interstellar journey Exocomet Express: How to Hitch Intersellar Rides on Comets. Just came across that while searching for material for this answer, not sure how practical it is.
See also:
In short, very likely that many stars have Oort clouds. But so far not confirmed for sure that any star does, even our own, though we don't have any good alternative hypothesis to the Oort cloud to explain the distribution of comet orbits AFAIK.
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