Various recent iterative optimization methods require to compute the proximity operator of a sum of functions. We address this problem by proposing a new distributed algorithm for a sum of non-necessarily smooth convex functions composed with arbitrary linear operators. In our approach, each function is associated with a node of a graph, which communicates with its neighbors. Our algorithm relies on a primal-dual splitting strategy that avoids to invert any linear operator, thus making it suitable for processing high-dimensional datasets. The proposed algorithm has a wide array of applications in signal/image processing and machine learning and its convergence is established.