Since Edward Snowden leaked details of the National Security Agency’s electronic surveillance program two years ago, the controversy over privacy and domestic spying has crossed party lines, embarrassed senior officials and launched presidential campaigns. The political furor led to passage of the USA Freedom Act in early June. It will stop the NSA from collecting information on millions of American citizens’ phone calls.

A Pew survey in May showed bipartisan majority disapproval of the NSA program—including 56% of Republicans,...