In this paper, a fully decentralized local energy market based on peer-to-peer(P2P) trading is proposed for small-scale prosumers. In the proposed market, the prosumers are classified as buyers and sellers and can bilaterally engage in energy trading (P2P) with each other. The buyer prosumers are equipped with electrical storage and can participate in a demand response (DR) program while protecting their privacy. In addition to bilateral negotiating with the local sellers, these players can compensate for their energy deficiency from the upstream market as the retail market at hours without local generation. In this paper, the retail market price is assumed uncertain. Robust optimization is applied to model this uncertainty in the buyer prosumers model. The proposed decentralized robust optimization guarantees the solution’s existence for each realization of uncertainty components. Furthermore, it performs optimization to realize the hard worse case from uncertainty components. A fully decentralized approach known as the fast alternating direction method of multipliers (FADMM) is employed to solve the proposed decentralized robust problem. The proposed approach does not require third-party involvement as a supervisory node nor disclose the players’ private information. Numerical studies were carried out on a small distribution system with several prosumers. The numerical results suggested the operationality and applicability of the proposed decentralized robust framework and the decentralized solving method.