Like most public property , contentions can exist.
Often by others who don't know what you are doing ,or don't care what you are doing and are busybodies.
Peeking in school windows or trying doors is obviously going to attract negative attention.
Just metal detect like you are metal detecting and be mindful to replace plugs the best you can to not leave signs of activity .
Expect someone to call in suspicious activity.
How you act if the law shows up matters.
If you're scared and nervous it will show.
If you are comfortable detecting for the right reasons it will show.
Signage is key obviously. Going past a sign restricting access or detecting is obviously not supportive of you being on a site.
An overzealous l.e.o. can ask you to leave even if nothing prohibits your being on a site. Which brings us back to how you act.
When you don't know what brought the officer there ,(driving by , a neighboring home owner , or the mayor called; are very different things) your staying there depends on more than if it's public ,or a lack of rule regarding detecting.
Detecting like you are part of the community and schools history and interest are different that pillage and plunder...
So id detecting for hints of the past through relics vs. treasurehunting.
Not schools , but public places in Mi. I've detected I waved briefly and went back to detecting when a cop stopped to watch.
Will again.
Village president and grounds keepers and residents have stopped at times in one park to chat for varied reasons.
None have objected locally. Though one dingbat comes to mind elsewhere that had no better impact on a site...
The police have all just watched a while and moved on.
But I do expect an approach and questions eventually ,which is fine.
I'll be polite and if asked to leave I will. But why I'd be asked to leave would be my question.
I vote ,have fun.