Denmark has been rocked by a surveillance scandal in which private citizens’ data was allegedly collected by military intelligence and then shared with foreign powers.

This week, Danes learned that their Defense Intelligence Service, known locally as FE, may have repeatedly broken the law by misleading its watchdog about spying on private individuals. The revelations, brought forward by a whistle-blower, have already resulted in several high-level dismissals at the agency.

At first, few details were provided to the public. But Denmark’s state broadcaster has since revealed that FE shared the private data it obtained with the National Security Agency in...