Similarity search finds objects that are similar to a given query object based on a similarity metric. As the amount and variety of data continue to grow, similarity search in metric spaces has gained significant attention. Metric spaces can accommodate any type of data and support flexible distance metrics, making similarity search in metric spaces beneficial for many real-world applications, such as multimedia retrieval, personalized recommendation, trajectory analytics, data mining, decision planning, and distributed servers. However, existing studies mostly focus on indexing metric spaces on a single machine, which faces efficiency and scalability limitations with increasing data volume and query amount. Recent advancements in similarity search turn towards distributed methods, while they face challenges including inefficient local data management, unbalanced workload, and low concurrent search efficiency. To this end, we propose DIMS, an efficient Distributed Index for similarity search in Metric Spaces. First, we design a novel three-stage heterogeneous partition to achieve workload balance. Then, we present an effective three-stage indexing structure to efficiently manage objects. We also develop concurrent search methods with filtering and validation techniques that support efficient distributed similarity search. Additionally, we devise a cost-based optimization model to balance communication and computation cost. Extensive experiments demonstrate that DIMS significantly outperforms existing distributed similarity search approaches.