Let's see - just quickly . . . .

Old Michigan City could be considered privately owned, but is open as a museum
Big Bay is a B&B
East Brother is a B&B
Two Harbors is a B&B and museum
Cold Spring Harbor is privately owned
Baliey's Harbor Range Lights are, I believe, privately owned by a group of some type.
St. Joseph Point FL - isn't this a private residence?

I'm sure there are others, and not all these are so private that they are out of sight/view.