In all, nearly a mile of East Hampton’s oceanfront shoreline could become off limits to the public if parallel lawsuits brought by a group of property owners prevail. There is a great risk that stretches of beach in Amagansett and on Napeague would be, in effect, privatized after centuries of being open to all who wished to pass. The property owners are claiming that a 19th-century sale by the East Hampton Town Trustees reserved for them alone the use of the area above the high-tide line. If they win in court, everything from beach driving to birdwatching could be blocked when the tide is up.