In microgrids, the integration of distributed energy resources (DERs) in the residential sector can improve power reliability, and potentially reduce power demands and carbon emissions. Improving the utilization of renewable energy in households is a critical challenge for DERs. In this regard, renewable power sharing is one of the possible solutions to tackle this problem. Even though this solution has attracted significant attention recently, most of the proposed power sharing frameworks focus more on centralized schemes. In contrast, in this paper, the performance of a proposed distributed power sharing framework is investigated. The problem is formulated as a repeated game between households in a microgrid. In this game, each household decides to cooperate and borrow/lend some amount of renewable power from/to a neighboring household, or to defect and purchase the entire demands from the main grid based on a payoff function. The Nash equilibrium of this game is characterized and the effect of the strategies taken by the households on the system is analyzed. We conduct an extensive evaluation using real demand data from 12 households of different sizes and power consumption profiles in Stockholm. Numerical results indicate that cooperation is beneficial from both an economical and environmental perspective and that households can achieve cost savings up to 20 %.