In this paper, we propose a decentralized distributed algorithm with stochastic communication among nodes, building on a sampling method called "edge sampling''. Such a sampling algorithm allows us to avoid the heavy peer-to-peer communication cost when combining neighboring weights on dense networks while still maintains a comparable convergence rate. In particular, we quantitatively analyze its theoretical convergence properties, as well as the optimal sampling rate over the underlying network. When compared with previous methods, our solution is shown to be unbiased, communication-efficient and suffers from lower sampling variances. These theoretical findings are validated by both numerical experiments on the mixing rates of Markov Chains and distributed machine learning problems.