I think the "private beach" in Orient refers to Club O section. There was a previous post about the special disposition to allow total nudity. Sorry but I don't remember exactly what it was.

La Samana has the right to a portion of Baie Longue and can keep people away. Nettle Bay has villas that purchase privacy in front of these villas.

Grand Case Bach Club has a portion of Grand Case beach that is posted "residents & guest only" .

But I don't know what is deemed private in Anse Marcel.