Of course, for this, you need a bit of determinism, and some games need to synchronize other parameters, which can lead to "security" concerns: bullet spread, hit detection and such for FPSs, for example.
This can also lead to some hilarious de-sync issues when a player tries to cheat on its local game instance. By altering its local state, the AI is effectively desynchronized, and the players can observe different outcomes on their local instances. This is often used as a punishment for cheating.
The alternative (for serverless, p2p games) is to have a central host, but it might take a lot of computing resources, and the host is then generally free to cheat.
One side effect of having deterministic AIs synced over the lobby is that the code needs to have the exact same behavior, down to the rounding errors. This can dramatically increase the complexity of cross-platform multiplayer games, and usually requires the exact same binary to be used by every client.
I think that Civilization and Sins of a solar empire use this kind of distributed AI scheme (as well as most games that don't have cross platform multiplayer, and those where you can experience de-syncs).