In this paper, we consider the problem of estimating a sparse signal over a network. The main interest is to save communication resource for information exchange over the network and hence reduce processing time. With this aim, we develop a distributed learning algorithm where each node of the network uses a locally optimized convex optimization based algorithm. The nodes iteratively exchange their signal estimates over the network to refine the local estimates. The convex cost is constructed to promote sparsity as well as to include influence of estimates from the neighboring nodes. We provide a restricted isometry property (RIP)-based theoretical guarantee on the estimation quality of the proposed algorithm. Using simulations, we show that the algorithm provides competitive performance vis-a-vis a globally optimum distributed LASSO algorithm, both in convergence speed and estimation error.