Top U.S. intelligence officials gathered in the White House Situation Room in March to debate a controversial proposal. Counterterrorism officials wanted to create a government dragnet, sweeping up millions of records about U.S. citizens—even people suspected of no crime.

Not everyone was on board. "This is a sea change in the way that the government interacts with the general public," Mary Ellen Callahan, chief privacy officer of the Department of Homeland Security, argued in...