Roughly 300 miles off the coast of South Florida is a string of mostly uninhabited small islands amid clear blue water and coral reefs on a limestone plateau. There's virtually no nightlife, no major airports, no celebrity chef restaurants. There are also no neighbors. That remoteness is attracting a rarefied group of wealthy entertainers and name-brand CEOs. In the world of high-end real estate, privacy is an increasingly prized amenity.

Islands in Exuma Cays, a 365-island archipelago in the Bahamas, have recently sold to...