The paper you posted seems to imply that such boundaries already exists (though I'm not a lawyer, it's probably not so simple). I see nothing that indicates school will actually do something about anything that's not criminal action or abusive behavior. If it does overstep it's competences, then then it's off to court.
Our justice system isn't perfect, but just monitoring everyone all the time until they inevitably break the law so we can hammer them for every mistake isn't the answer.
"Inevitably" again. No, it's not inevitable that one will break the law. There are people who carry on with their lives without breaking a single law in their lifetime. In fact, most American people never ran afoul of the law. So, unless you're actually talking about an actual certainty (i.e. death and taxes), stop with the "inevitably". Most people have no business breaking the law.
While making school system responsible for this might not be the best idea, who else would you task with that? The NSA? They're not exactly made for taking care of the children (though given their overinflated funding, they could certainly afford that). A new organization? As if there weren't enough of them already. Schools do have some experience in this matter, so they seem the least problematic choice here. They're underfunded, yes, but perhaps the answer is to increase their funding. As for where to get it? I'm pretty sure NSA and TSA don't need all this money they're getting, especially the latter.
Everyone has broken the law at some point. We have some really, really ridiculous, overly specific, and laughably outdated laws on the books. In fact, if you look hard enough, you could probably find examples of most people breaking some kind of law every single day. People who go through life with clean records simply either haven't been caught breaking laws or just haven't broken any that are actually enforced. I've never even been pulled over for a traffic violation, but that doesn't mean I've never done anything wrong.
As to who should be monitoring everyone all the time, the correct answer is no one, because there's a critical point where far-reaching power invites far more problems than it solves. At some level, I believe we have to accept that humanity is not perfect, society is not perfect, and no matter how hard we try, some things are going to slip through the cracks. Making everyone into a criminal because they said something unwise once on Facebook is not the answer to...well, whatever sufficiently nebulous problem this surveillance is supposed to solve. Schools becoming big brother because of some generic "think of the children" tagline is unnecessarily alarmist. If this is about bullying, the schools already have the power to cut down on it, and they have for quite some time. The problem is that they're allowing bad parents to interfere with the process. At some point bullying also can run afoul of the law, and that's where the school's authority ends and the police should be informed. And again, if it's for criminal behaviour, that's what the police are for.
If all the schools ever do is inform the proper authorities of possible criminal activity, then there's nothing wrong with that. But if you think they'll stop there, you haven't had to deal with any school administrators. There are already plenty of stories about punishments for non-criminal activity. Do you remember the story about the school laptops that had webcams that could be activated remotely, and that a student was disciplined for "improper behaviour" that took place in his own room? The FBI had to step in and investigate the school district. The really disgusting thing is that using webcams and keyloggers to spy on students wasn't found to be illegal, because it should be. I know that this thread was referring more to public Facebook posts and the like, but again, we already have a system in place for that. If you see illegal behaviour, harassment, or bullying on Facebook, report it to the moderators and the police, if necessary. Educating people about their rights, responsibilities, and obligations is the best possible solution - which, ironically, is part of what the schools are SUPPOSED to be doing, not spying on students.
EDIT: I want to clarify that I'm not trying to come across as confrontational - I don't disagree that the schools have problems, bullying is a problem, harassment is a problem, etc. And, I agree there's really nothing inherently wrong about searching public information. I just don't think turning the school district into the NSA is the right solution.