Open Feint has a notion of challenges -- where a player can finish a game and then send a challenge to another user (like a "beat my score"). The app passes along whatever game data it needs to the OF servers and then when the challenge is accepted, the info is downloaded and the challenge is started.
Is there something similar in Game Center? I've been looking through their docs, but it seems they are more interested in live multiplayer scenarios.
Andrew
To do this, you need to have your own server, and you should use Push Notifications. It's not (yet) possible with Game Center at the time of writing (4 november 2010). However, iOS 4.2 is coming soon and it has many new things. :)