An idea I got from Steve Jackson Games's original Illuminati game....Not the card game, the real one.

In the box, as I recall, were two full sets of credentials for two out of five different secret societies, one false, one true. Also, there was a couple of clues for a logic puzzle of the "Who has the Zebra?" type. Hence this game.

I give you the basic idea for a group of 20, feel free to add and subtract as you will.
Prepare 20 boxes, similar to the one given above. Each box will contain 2 sets of credentials, and two clues for a logic puzzle.The only difference between the false and the true credentials are the logic puzzle clues: for instance, The (true) Knights of Demosthenes will have "The woman in the pink house knows that the lady with the cat hates tea." while someone with a false Knights of Demosthenes credential will have another clue altogether. However the two Knights won't have the same false credentials: say, one will be a (false) Son of Neptune, and the other will be a (false) Assassin 's Guildmaster. Each of these sets of credentials should have a good deal of backstory, such as "The Sons of Neptune were established in 1824, among merchant sailors in New Bedford, MA. They include many fishermen and commercial ship's captains, some Navy personnel, and a few yachtsmen among their membership rolls." Mix all these boxes up, so that no one knows who has what.

Over a weekend, the object is to a) find your partner, and b) solve the puzzle BAMN.

Some notes on strategy:
Talk, talk, talk. Use your backstory to help you. If you need to find another Knight of Demosthenes, talk about knights, public speaking, Yale University...
Also, listen. Maybe that guy discussing Middle Eastern affairs is an Assassin, maybe not. But there's a good chance he's trying to find his partner if he is.
Unless you know the other guy is not your partner, don't be rude. Maybe they've infiltrated a group you need to solve the puzzle.
Good luck!