Smart contract effects

I have a question about smart contracts. Given that each node on the network runs the code attached to a contract independently, wouldn't the effect of the code be executed multiple times? For instance, if the code transfers coins from one address to another, wouldn't each node initiate a separate transfer? Surely, some mechanism is in place to prevent this from happening, but does anyone know how it works and what the policy is? Which node is allowed to enact the change? Also, how is a race condition prevented?