Consider such situations: user clones a repo from an organization github.com/org/repo.
Is there a way how info about user's privileges (read, write, etc.) can be gotten via gitpython package?
I'm writing a school app, which will allow automatically commit and push to repo with one button (but app should show this button only to user's with privileges).
No, you can not. Git permissions are ruled by server, so you can not get these rights locally except by trial and error (which I assume you don't want to do). Also, you can have multiple remotes including local ones.
In the particular case of Github, you can get permissions via GithubAPI