In general, training requires far more neural networks requires far more computational power than using them does. However, AlphaGo combines neural networks with a traditional Monte Carlo search. This means that you can improve the performance of a trained AlphaGo by just giving it more processing power. Indeed, with a sufficient amount of processing power, AlphaGo would converge to fully optimal play.
It's not really the traditional Monte Carlo search; IIRC they're using UCT weighting for MCTS. Apologies if that's what you meant, but I think to most people "traditional Monte Carlo" means something different (probably uniform depth charges).
