WASHINGTON—Bassam Barabandi, a diplomat in the Syrian embassy here, was leaving Ramadan prayers at a mosque late one night two years ago when he ran into an opponent of the Syrian regime. Mr. Barabandi knew the man slightly and offered him a ride home.

The chance conversation that followed transformed the diplomat's life. It also altered the fortunes of scores of Syrians working to overthrow Syrian President Bashar al-Assad.

"I...