Distributed compressive sensing (DCS) concerns the reconstruction of multiple sensor signals with reduced numbers of measurements, which exploits both intra- and inter-signal correlations. In this paper, we propose a novel Bayesian DCS algorithm based on variational Bayesian inference. The proposed algorithm decouples the common component, that characterizes inter-signal correlation, from innovation components, that represent intra-signal correlation. Such an operation results in a computational complexity of reconstruction which is linear with the number of signals. The superior performance of the algorithm, in terms of the computing time and reconstruction quality, is demonstrated by numerical simulations in comparison with other existing reconstruction methods.