Communities across the country are grappling with a surge in drone use that’s raising safety and privacy concerns—and thorny legal questions—about a slice of sky officials have largely disregarded.

State and local police say complaints are soaring about drones flying above homes, crowds and crime scenes. At least 17 states, meanwhile, have passed laws to restrict how law enforcement and private citizens use the devices—preemptive policies that many drone users say are heavy-handed. And despite federal regulators’ stance that...