A critical issue not made clear here is whether the postings involve any school owned facility. That is, if someone uses a school computer, school internet service, or school website, then the school district has every right and a duty to set standards and monitor traffic. Kids can be quite nasty on the Internet and the school could be held liable for malicious postings that hurt another kid. The school could also be liable if a kid gets hurt as a result of posting too much personal data.
On the other hand, if school facilities are not involved in any way, then it is not the school's business. If two student get into a fight at the movies or the mall on a weekend, then it is not the school's business.
I can understand the school's concern for inappropriate posting, especially those that might bring risk to a student (e.g. publicly posting a provacative personal picture and personal info, or malicious gossip about other kids). But the most the school can do for non school facilities IMHO is notify the parents and advise such postings aren't a good idea.
If I were a school administrator, the last thing I'd want to do is worry about what my kids did outside of school hours and property*. If I were a student, I'd feel my rights were being violated and the school is snooping on me.
*Perhaps if a student was arrested for a serious crime the school has a right to be involved.