We consider a model for repeated stochastic matching where compatibility is probabilistic, is realized the first time agents are matched, and persists in the future. Such a model has applications in the gig economy, kidney exchange, and mentorship matching. We ask whether adecentralized matching process can approximate the optimal online algorithm. In particular, we consider a decentralizedstable matching process where agents match with the most compatible partner who does not prefer matching with someone else, and known compatible pairs continue matching in all future rounds. We demonstrate that the above process provides a 0.316-approximation to the optimal online algorithm for matching on general graphs. We also provide a 1/7-approximation for many-to-one bipartite matching, a 1/11-approximation for capacitated matching on general graphs, and a 1/2k-approximation for forming teams of up to k agents. Our results rely on a novel coupling argument that decomposes the successful edges of the optimal online algorithm in terms of their round-by-round comparison with stable matching.