When I was co-editor of my high school's student newspaper, every issue was subject to administrative review. Our principal literally read every issue prior to publication and told us what could and couldn't be run.
Of course, a school intent on interfering with a newspaper is probably not going to make that part of the curriculum.
There is a New Voices movement in multiple states to strengthen students' rights regarding school newspapers: http://www.cjr.org/united_states_project/new_voices_campaign...
No different from any "real" newspaper.
But those publishers aren't subject to the First Amendment, because they're not government-run.
In the case of a school newspaper, however, the school is run by the state, and so First Amendment issues definitely do come into play.
They have a much harder job than it might appear at first blush -- even the slightest hint of repressing a valid story turns a very harsh spotlight on them.