Community search is important in social network analysis. For a given vertex in a graph, the goal is to find the best community the vertex belongs to. Intuitively, the best community for a given vertex should be in the vicinity of the vertex. However, existing solutions use \emph{global search} to find the best community. These algorithms, although straight-forward, are very costly, as all vertices in the graph may need to be visited. In this paper, we propose a \emph{local search} strategy, which searches in the neighborhood of a vertex to find the best community for the vertex. We show that, because the minimum degree measure used to evaluate the goodness of a community is not \emph{monotonic}, designing efficient local search solutions is a very challenging task. We present theories and algorithms of local search to address this challenge. The efficiency of our local search strategy is verified by extensive experiments on both synthetic networks and a variety of real networks with millions of nodes.