They are ALL private. And in recent years the Portsmouth Harbormaster has made the rounds on busy weekends ensuring that the people on the moorings are, in fact, the owners. I've personally watched them do this and then do a safety inspection on boats they've made leave after it's dropped the mooring. It is a $100 fine if they decide to ticket you.
Anchoring in Potter's is very easy. Holding is mud and it's easy to set. There is room around the Shell island inside the pond which is awash at high tide. Just be careful. There is also room just to the right of the last Red nun as you enter as well as to the left, along the shore althought that is a bit exposed to boat wakes.