"Send me your paper when you're done with it, I'd love to read it" I noticed that at least in my field, sometimes professors say to students, "send me the [research] paper when you're done with it, I'd love to read it [and give comments]". That is usually taken as a compliment that the professor is interested in the student's work, especially if the professor is famous.

Is it appropriate for students to say that to professors, especially if they're genuinely interested in the professor's research? Or could it be seen as pretentious?

For context, I'm a social scientist, and my discipline is not grant-driven at all, so it is very common for students to not work with their advisors on particular projects (e.g. they work with other graduate students, or other professors who are not their advisor)
It's not practical or appropriate for a student to impose on the professor a burden of remembering to send them something at some undefined future time.

More generally, any request of a professor of the form "Do X at time Y" where Y is more than 5 minutes away, and even more so a request such as "Do X after you've completed task Z", is 99% likely to be ignored (and potentially be perceived as quite rude) unless the person making the request is the professor's superior, good friend, highly respected colleague, or funding agency. (Even in those scenarios, there is still a non-negligible probability that the request will be ignored!)

Of course, the student is welcome to keep independent watch of the professor's publications page or preprint archive where they post their papers, read the paper when it is posted there, and send them any comments they like; I don't see anything inappropriate about that, and don't see a problem with asking a question such as "is it okay if I send you any comments I have after reading your paper once it's posted online?" (although the question is superfluous -- I would just assume that it's okay and that no special permission needs to be granted).

To clarify, the issue is not the pretentiousness of the request, but the fact that it shows a rather extreme ignorance of, and lack of sensitivity to, how busy professors are and how hard it is to reliably schedule tasks that are dependent on other tasks being completed.