Schools should be able to access the social media activities of students when there is a compelling interest in maintaining the safety and welfare of the student body. In addition, any time a student is using the school's network to access their accounts, the school should have the right to monitor their activity with a legitimate need. There should be restrictions on who has the right and ability to look at the accounts to avoid the situation where anyone can find anything on anyone.
The personal profiles should not be monitored by the schools they attend. This is a student's personal space for them to express themselves. If it is a public university, and there is nothing on there that is not threatening to others, than no one should be able to dictate what is posted.