Low voltage distribution networks were not designed to support massive deployment of distributed energy resources (DER) such as solar panels, which is currently hindering the Energy Transition. Recent research contributions have shown that local energy markets improve the capacity of distribution grids to host DER. In parallel, distributed models were created to deal with the control of batteries in presence of stochastic demand and production, as well as variable electricity prices. The combination of the two techniques has received little attention until now, from both the literature and the industry. In this paper we extend the traditional approach to sequential stochastic decision processes by also modeling the interaction with the neighborhood in addition to the utility. The model is then solved using reinforcement learning techniques. For the local energy market we use MUDA: a strategy-proof multi-unit double auction. The performance of the proposed system is evaluated through simulations, demonstrating its capacity to effectively decrease the overall exchange of energy with the grid and the monetary cost for users.