We've got some really big apartment complexes in my city .... so big, that they have their own little play-lot, sandbox, etc... And I highly doubt anyone is going to know a tennant from a non-tennant. But sure ... knock yourself out and try to get permission if you want.
For example: I live in a private gated community. And we have our own softball field, playground, etc.... On any given day, there'll be random people using the park. Including full-scale games, because residents bring over their school sports practice leagues, etc... There's utterly no way to know who's from actually living there, to who's a guest, to who ........ simply drove in.
Now since I attend our H.O.A. meetings and know board members, I can just envision someone coming in to request permission to metal detect. Odds are, they'd be told "no". (afterall, why do *we* want to be bothered? How do *we* know that every other "yahoo" won't start showing up "looking for gold", blah blah). But as I say, if you just went, ..... no one would know you from Adam. In fact, when I first moved here, I did, in fact, hit our own park (before anyone even knew me yet). I was not asked for anything, no one approached me, etc... (in fact, I was the only one in the entire park as it was dusk and cold late that mid-week afternoon, etc....). So I could have just been Joe-Blow and not have been a resident.
But granted, there's about 100 homes in our HOA, so ..... you've got to use a little common sense on where you hit, how you conduct yourself, etc...