There is supposedly a record of who has visas and when those visas expire. It isn't all that complicated to find out whether those people are still here and, if they are still here after their visas have expired, to deport them. There is no need for any kind of Gestapo-like organization.

__________________
A panda walks into a cafe. He orders a sandwich, eats it, then draws a gun and fires two shots in the air.

"Why?" asks the confused waiter, as the panda makes toward the exit. The panda produces a badly punctuated wildlife manual and tosses it over his shoulder.

"I'm a panda," he says at the door. "Look it up."

The waiter turns to the relevant entry and, sure enough, finds an explanation.

"Panda. Large black-and-white bear-like mammal, native to China. Eats, shoots and leaves."