If it's a public school, then it's a public school. No permission needed. I suspect if some school personell ever took issue with anyone detecting, it was not necessarily because "permission was needed", but rather, that they though you might leave holes. If it is open for other after-school usage, like to use the basketball hoops, or jog the track, or whatever, then think of yourself as no different than anyone else.
Here in CA, a lot of city's schools have put up fences in recent years, but there is usually still a pedestrian gate opening, that's not fully shut. I heard that the reason for these fences and "permission to pass revokable ... blah blah" signs, was because of lawsuits like the following true story:
In some Los Angeles suburb, a young married couple took their infant kid to the local school yard playground equipment. They were pushing the kid down the slide, when he fell over the side and was permanently injured

I don't know the age of the infant, but he was still in diapers, however young that is! The parents sued the school district, and
WON! Now I don't know what kind of sense it takes for parents of a kid that young, to go pushing him down a slide, but seems to me THEY should be held responsible for the safety of their kid,
not the school district.
The person telling me this story says that one day they showed up to detect their favorite schools, and found these fences and signs. They then found out that this lawsuit was the reason why. Nothing to do with metal detecting. People still use the park, by going through the turnstyles or narrow openings where the students go through. I guess it (the signs or fences) is enough that if, in the future, some moron did injure himself, the schools could say "well he shouldn't have been there", etc... The truth is no one usually cares, but is just protecting themselves from litigation. The average schools in my area still has ball games, kite flyers, dog walkers, etc... after school hours. So I will not consider myself any different from them. To ask, you'll only get the obligatory "no", when in fact, the odds are, no one really cared.