Mchili

I read that our Oort cloud could extend up to halfway to Proxima Centuri. Is it reasonable to assume that this star could also have its own equivalent of an Oort cloud and if so, could the two clouds merge at some point? If true this leads to my main question. Would it be possible to have a comet that had an orbit that included both suns? At least until stellar drift upset things anyway. I am not an astronomer and am not that proficient with orbital mechanics so apologies if these are stupid questions.