A growing number of Americans are getting permission to carry firearms in public—and under their clothes—a development that has sparked concern among some law-enforcement authorities.

Applications for "concealed-carry" permits are soaring in many states, some of which recently eased permit requirements. The numbers are driven in part by concern that renewed gun-control efforts soon could constrain access to weapons, along with heightened interest in self-defense in the wake of mass killings in Newtown, Conn., and...