Federal prosecutors alleged 11 people were spies living secret lives in American communities, from Seattle to Washington D.C., sent years ago to infiltrate U.S. society and glean its secrets.

In an extensive and bizarre affidavit whose details echoed Cold War spy thrillers, the Federal Bureau of Investigation claimed the alleged spies were sent here by the Russian overseas intelligence service known as the SVR—the successor to the Soviet KGB—as early as the mid-1990s, and were provided with training in language as well as...