NANTUCKET, Mass.—Local voters here will decide Tuesday just what kind of place this small island should be.

They have two measures to decide on Election Day: one would put tight limits on rental housing, which boosters say would preserve more real estate for islanders. A competing measure ensures Nantucket stays open to both long- and short-term renters, who some property owners and small businesses say are vital for the local economy.

The dueling proposals encapsulate a long-running and contentious struggle that has fractured relationships, prompted court battles and sparked feuds among millionaires and billionaires. Voters already have batted down eight proposals in the past five years that would restrict short-term rentals, but the fight keeps intensifying. This one involves an especially consequential question for out-of-towners: whether weekenders are welcome.