The paper deals with distributed learning of Nash equilibria in games with a large number of players. The classical fictitious play (FP) algorithm is impractical in large games due to demanding communication requirements and high computational complexity. A variant of FP is presented that aims to mitigate both issues. Complexity is mitigated by use of a computationally efficient Monte-Carlo based best response rule. Demanding communication problems are mitigated by implementing the algorithm in a network-based distributed setting, in which player-to-player communication is restricted to local subsets of neighboring players as determined by a (possibly sparse, but connected) preassigned communication graph. Results are demonstrated via a simulation example.