I hunt old-town demolition sites all the time, with no hesitation. I just wait for the workers to cut out at 5pm. They can be anything from 1) no fence at all 2) a simple yellow ribbon and some cones (like for sidewalk demolition), 3) a sloppy joke of a fence that has a gate that swings wide open, no signs, etc... to 4) full-blown tight fencing. So it all just depends on what level of scrutiny you're talking about. But go figure, there are people on the forum that won't even go to a public SAND BOX without "permission", so you're asking a loaded question.
You gave, as an example, a public shool demo. in your city. Using that as an example: they tore down and out portions of our old high school in my city, years ago. In the process, they ripped out all the deep thick crab grass from the inner court, where ....... for 80+ years, hundreds of kids each day sat to eat their lunch. This particular site did have fencing, but I would categorize it as the "joke" kind that you merely walk right through wide gaps, or un-do the bailing wire "latch", etc... We hunted it for 2 straight weeks, in full view of Main St. traffic, with after-school staff (a portion of the school was left operable) just passing by with nothing more than a friendly wave. Never had a problem. Of course, that was with a public school, so someone might argue that a privately owned site would be different

We found hundreds of silver coins, a silver dollar, a 2-cent piece, etc.... I suppose if we were to have asked at some bureaucrat's office across town, or to have asked the const. co, they might have had to pass it through their legal channels, take a vote, and say "no".
That is just one example. The only time I ever ask for demolition sites, is if there is just no other possible way to access it (like if there are roaming guards, or whatever)