This would seem to vary greatly based on locality.
Yes, sorry, I didn't mean to imply that this was true everywhere. Just to state that having a full-time nurse is not a huge exception. I'm in a small town (about 20k) which has 11 public schools (8 Elementary, 2 Middle, 1 High). Judging from their web page, they have 11 nurses and a nurse supervisor, so it appears to be one per school. Where my children attend an Elementary. Middle, and High school together, and there are three nurses in total (2 for Elementary, 1 for Middle, 1 for High). One of the schools does not have a nurse listed; I don't know where that one is, but I suspect it's co-located with another of the schools that does have a nurse.
It will vary by district and funding level. Rural and lower-income areas would be much less likely to have full-time nurses, and would be tend to part-time nurses or the sharing that others indicated.
ETA: I see some additional info has been posted. And blutoski also has a point, I suspect the policy for our school is in place because we do have two full time nurses. At other schools (including one I went to when younger, about *mumble mumble* years ago), medicines were kept in the prinicpals office. The goal is to have an adult involved to make sure medications are taken safely and correctly. I could see loosening the standards at a point (say, at a certain grade level or age they can carry them themselves as long as there's a note from the parent), but for younger children I most definately want an adult involved.
And I simply can't agree with an open policy on OTC medications. I don't want the school, a place where I have no insight and no oversight, to allow my children to take whatever OTC meds they want wihtout my input or knowledge. An "open" OTC policy would be specifically taking this choice away from me as a parent. It would remove my ability to have any say in what my child was taking or how. In the current system, we get a one-page report from the nurse when one of our children needs medicition, saying when they came in, what they compalined of/needed, what they were given, and so forth. Basically, the school policies concerning OTC meds are to insure two things:
1. Of course, so that the school can cover itself from liability issues by trying to insure the safety of the students.
2. So the parents are informed and aware of what their children are taking, and the school does not provide an environment that removes that oversight.
Now, since if I don't post it every few posts it seems someone gets the wrong idea about my position (

), I am absolutely against the one-punishment-fits-all zero tolerance BS. I think the punishment should fit the crime. Use of OTC meds against policy, assuming it isn't abuse, should have a mild punishment (maybe even simply confiscation of the meds and notifying the parents for a first offense, moving up with repeated offenses, same as most other rules).