Local community detection aims to find a set of densely-connected nodes containing given query nodes. Most existing local community detection methods are designed for a single network. However, a single network can be noisy and incomplete. Multiple networks are more informative in real-world applications. There are multiple types of nodes and multiple types of node proximities. Complementary information from different networks helps to improve detection accuracy. In this paper, we propose a novel RWM (Random Walk in Multiple networks) model to find relevant local communities in all networks for a given query node set from one network. RWM sends a random walker in each network to obtain the local proximity w.r.t. the query nodes (i.e., node visiting probabilities). Walkers with similar visiting probabilities reinforce each other. They restrict the probability propagation around the query nodes to identify relevant subgraphs in each network and disregard irrelevant parts. We provide rigorous theoretical foundations for RWM and develop two speeding-up strategies with performance guarantees. Comprehensive experiments are conducted on synthetic and real-world datasets to evaluate the effectiveness and efficiency of RWM.