In the risque version, the players still throw out random words, but at some point, one of the players (usually the one who proposed that they play the game) tells the other one a deep, dark secret. This secret may or may not involve the other person or their life, and the other person might come back with a secret of their own. The person with the deepest, darkest secret can "win," even though the random game isn't "winnable," per se.
The risque version is usually more common in forms of communication other than face-to-face, such as texting or IMing. The fact that the players are miles away from each other temporarily saves the secret-teller from a potential ass-kicking. The clean version can be played through just about any means.
Both versions can go on for theoretically any amount of time, although the risque version usually ends quickly due to the awkwardness/anger caused by the secret-telling.
Long story short, if someone asks you to play the random game.....they probably have a confession to make. I suggest you either brace yourself or decline the offer.
Jack: Hey bro, let's play the random game.
Tom: Okay, dude.
Jack: I fucked your sister the other night.
Tom: She fucked your brother that same day.
Tom: I win.
Molly: Let's play the random game!
Molly: I walked in on your husband doing it with a dude.
The clean version:
Billy: Let's play the random game!
(participants continue to say random words)