I'm not a big fan of private islands, but Princess Cays on Eleuthera Island in the Bahamas was lovely. My husband and I rented a clam-shell for the day, and sat on the beach at the quiet end of the island (across the bridge from the tender pier-- to the right).
One thing I will commend Princess for, were the number of luncheon buffet lines. There was never a line at our end either for drinks at the bar, or for lunch at the buffet. This was very well planned and executed.