I haven't yet been to Anse Marcel post Irma but will correct that in June. However, pre-Irma, there was a small sign as you entered that bringing your own food and beverage was not allowed. I don't recall a "private beach" sign at that time. Part of the reason I go there is for their food and beverage, which are excellent from all reports.
I don't remember a private beach sign either, but never really looked for one. We also enjoy Anse Marcel very much and go for the food and beverage. We can't do that every day, of course, but it is one of our favorite beach restaurants.