I read the OP's question but skipped the rest of the replies. For what I understand of public schools here in Texas, told by an ISD security officer, it's up to the principal of the school as to whether you can detect it. To me, that sounds a little low on the totem pole but sorta makes sense since he/she is the one immediately in charge of the property. Now, it is paid for with public funds (ie: property taxes), but even though paid with public funds it is private property. It belongs, not even to the city, but to the ISD of the city. And they take the "independant" in ISD seriously.
I've been told to leave by security officers and police officers while hunting elementary schools, and I've been told it's okay by other security officers and other police officers. Some are interested in my finds (curious about the hobby) and others just boot me out. I always leave no trace, properly dispose of all the junk I dig out of the playground, report suspicious activity direct to a security officer of the ISD that gave me his card...if I see anything weird going on while hunting at night. To this this one specific security officer, I'm an extra set of eyes. To others, I'm the trespasser. So it just depends?
I've never been ticketed, I completely cooperate with whatever they say, and I pull out a bunch of junk for them that might possibly injure kids. Not long ago, I found a door key to the school marked "do not duplicate" and I called the security officer to turn it in. The officer that showed up to retrieve the key didn't like that I was there, but after giving him the key...he let me continue. So my luck has been mixed.
I'm not exactly sure the entity within the ISD that would need to be asked for permission to detect, but pretty sure the answer would eventually be "No" so I'm not even going to attempt to climb that pole. I'm take my luck playing dumb till it wears out. I also bring my youngest with me so he can play on the equipment while I detect if not too late in the evening, I think that helps soothe it over too. Never once been approached when he was there with me playing.