The typical approach for recovery of spatially correlated signals is regularized least squares with a coupled regularization term. In the Bayesian framework, this algorithm is seen as a maximum-a-posterior estimator whose postulated prior is proportional to the regularization term. In this paper, we study distributed sensing networks in which a set of spatially correlated signals are measured individually at separate terminals, but recovered jointly via a generic maximum-a-posterior estimator. Using the replica method, it is shown that the setting exhibits the decoupling property. For the case with jointly sparse signals, we invoke Bayesian inference and propose the "multi-dimensional soft thresholding" algorithm which is posed as a linear programming. Our investigations depict that the proposed algorithm outperforms the conventional $\ell_{2,1}$-norm regularized least squares scheme while enjoying a feasible computational complexity.