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We consider the problem of correlated data gathering in sensor networks with multiple sink nodes. The problem has two objectives. First, we would like to find a rate allocation on the correlated sensor nodes such that the data gathered by the sink nodes can reproduce the field of observation. Second, we would like to find a transmission structure on the network graph such that the total transmission energy consumed by the network is minimized. The existing solutions to this problem are impractical for deployment because they have not considered all of the following factors: (1) distributed implementation; (2) capacity and interference associated with the shared medium; and (3) realistic data correlation model. In this paper, we propose a new distributed framework to achieve minimum energy data gathering while considering these three factors. Based on a localized version of Slepian-Wolf coding, the problem is modeled as an optimization formulation with a distributed solution. The formulation is first relaxed with Lagrangian dualization and then solved with the subgradient algorithm. The algorithm is amenable to fully distributed implementations, which corresponds to the decentralized nature of sensor networks. To evaluate its effectiveness, we have conducted extensive simulations under a variety of network environments. The results indicate that the algorithm supports asynchronous network settings, sink mobility, and duty schedules.