A lot of the area is given over to hiking trails that intersect the private road. So there's a lot of plausible deniability with the possibility of missing the "official" signs.

I only met one other person walking in the private area (and not a single car), and he looked just as uncomfortable as I felt. I've learned the trick in these situations is to pretend you belong and assume the other person doesn't, then see how it plays out from there.