their?

"Their" is used as a possessive for "the user" (to avoid gender pronouns like his or hers).

"they've": they is a pronoun for "the user" and is a contraction of "they have".

Taka wrote:

Initially, I also took it that way, but then, grammatically, doesn't 'the user' have to be 'the users' instead?

If there's only one user you are referring to, then you can't say "the users". Assuming you are referring to something like Youtube, the user is probably a member, someone who has an account and the the one who uploaded the video. If you can't comment on their video, they probably don't allow anyone to write a comment.

If you want to avoid they and their you should use "he/she" and "his/her" instead, which would already be pretty ugly to read, let alone say it. If you never heard of "singular they", try searching this forum, there are several threads that deal with it. Or search the net, it you want, there's a lot of stuff out there too.