You call it Reversi, I’ll call it Othello, but whatever you call it, it’s a classic game. The latest version to hit the Marketplace is called Faceted Reversi and it has a simple description (that doesn’t do it justice): “Play reversi against other people, either on the same phone or on different pone. Play someone you know, or let us choose a random opponent.” The app usage shows that push notifications are used in the game for turn by turn notifications. From the screenshots, you can see that there’s also a chat feature built into the game.
It’s available for $2 and has a free trial.