It's good practice to learn on public networks that you can and probably will be monitored

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My school banned pretty much anything that you could use to send messages on, and had their own mailserver, that everyone knew was monitored. This allowed people to send messages to each other, but ensured they stuck by the rules. (of course the odd idiot got pulled up for something though)
If it was a case of schools effectively being able to be given backdoor access to the accounts it would definitely be wrong, but if it's monitoring communications on their own internet connection, well, it's to be expected imo. Most workplaces and schools have a monitoring policy, and you can always go somewhere else if you don't want your employer/school looking at your content.
There were certain things I used to bookmark and then go browse at home

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If it's a case that the bullying is that of an extent that warrants private account investigation, it would need to be done by the right person. I don't think most school employees would be the right people for the job - it would be better done by someone who doesn't even know the kids, as the private accounts may turf up stuff that really shouldn't be seen by people the kids know. How many teachers are going to go OTT over some spur-of-the-moment insult against them, that was only intended for friends? Could the teacher/staff stay impartial if they read about some personal tragedy that the kid has suffered outside of school?
If someone tried to discuss an issue with my life, where there was only one way of finding out about it, I'd be very pissed about it, even if they were trying to help.