Drone industry and law-enforcement officials are struggling to find common ground over expanding flights and protecting public safety, a debate thrust into the public spotlight by a reported assassination attempt on Venezuelan President Nicolás Maduro.

The Federal Aviation Administration is projecting a fourfold increase, from more than 110,000 currently, in the number of commercial drones flying in U.S. skies in the next five years. U.S. law-enforcement officials, however, want to delay widespread operations until reliable...