We propose a distributed Bayesian quickest change detection algorithm for sensor networks, based on a random gossip inter-sensor communication structure. Without a control or fusion center, each sensor executes its local change detection procedure in a parallel and distributed fashion, interacting with its neighbor sensors via random inter-sensor communications to propagate information. By modeling the information propagation dynamics in the network as a Markov process, two-layer large deviation analysis is presented to analyze the performance of the proposed algorithm. The first-layer analysis shows that the relation between the probability of false alarm and the conditional averaged detection delay satisfies the large deviation principle, implying that the probability of false alarm according to a rare event decays to zero at an exponentially fast rate when the conditional averaged detection decay increases, where the Kullback-Leibler information number is established as a crucial factor. The second-layer analysis shows that the probability of the rare event that not all observations are available at a sensor decays to zero at an exponentially fast rate when the averaged number of communications increases, where the large deviation upper and lower bounds for this rate are also derived, based on which we show that the performance of the distributed algorithm converges exponentially fast to that of the centralized one, by proving that the defined distributed Kullback-Leibler information number converges to the centralized Kullback-Leibler information number.