Community-based energy initiatives, such as local energy markets (LEMs), are gaining attention as a means to foster a more equitable and participatory energy transition. To fulfill this potential, local communities require robust energy-sharing mechanisms that ensure the fair distribution of financial benefits among participants. This paper presents a novel algorithm for virtual energy distribution in a Peer-To-Community (P2C) local energy market. The algorithm first determines the optimal number of participants in each time period of the LEM, based on their current energy supplier tariffs, to ensure benefits for all participants. It then applies an egalitarian energy-distribution method, called the Glass-Filling algorithm, to allocate locally produced energy within the community. Using real-world data from a UK trial involving 200 households, half of which are prosumers with solar PV and residential batteries, we demonstrate that the proposed mechanism improves equity in energy sharing compared with a benchmark double-Auction mechanism. The proposed market mechanism yields an average annual bill reduction of 9.9% for producers and 5% for consumers.