Am I likely to get permission at a Private Airport? I'm planning a Labor Day flight to a relative's house, and the nearest airport is a Private Airport that looks to be part of a flying community (multiple houses all with hangars directly off the small runway).

I have no experience with private airports.
If I ask nicely in advance, am I likely to get permission to use it?
Or are they likely to be protective of their privacy and airport?

I realize any answer can't apply in all circumstances, but I'm asking if its worth a phone call and conversation? or if I should just go directly to the nearest public airport?
Can't hurt to ask!
I don't know the specifics of the situation, but I would be willing to bet that your chances are better than you think.  If it was one farmer with a grass strip and no accommodations for visiting aircraft you might be out of luck, but from what you describe it doesn't sound like one extra airplane on the ground for a few hours is going to make that much of a difference.
In fact, it might be that Fred (or whoever) will also be visiting friends for Labor Day and might be willing to let you use his ramp.
They can't say yes if you don't ask.