Reconstructing compressed sensing signals involves solving an optimization problem. An example is Basis Pursuit (BP) , which is applicable only in noise-free scenarios. In noisy scenarios, either the Basis Pursuit Denoising (BPDN)  or the Noise-Aware BP (NABP)  can be used. Consider a distributed scenario where the dictionary matrix and the vector of observations are spread over the nodes of a network. We solve the following open problem: design distributed algorithms that solve BPDN with a column partition, i.e., when each node knows only some columns of the dictionary matrix, and that solve NABP with a row partition, i.e., when each node knows only some rows of the dictionary matrix and the corresponding observations. Our approach manipulates these problems so that a recent general-purpose algorithm for distributed optimization can be applied.