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The smart grid incentivizes distributed agents with local generation (e.g., smart homes, and microgrids) to establish multi-agent systems for enhanced reliability and energy consumption efficiency. Distributed energy trading has emerged as one of the most important multi-agent systems on the power grid by enabling agents to sell their excessive local energy to each other or back to the grid. However, it requests all the agents to disclose their sensitive data (e.g., each agent's fine-grained local generation and demand load). In this paper, to the best of our knowledge, we propose the first privacy preserving distributed energy trading framework, Private Energy Market (PEM), in which all the agents privately compute an optimal price for their trading (ensured by a Nash Equilibrium), and allocate pairwise energy trading amounts without disclosing sensitive data (via novel cryptographic protocols). Specifically, we model the trading problem as a non-cooperative Stackelberg game for all the agents (i.e., buyers and sellers) to determine the optimal price, and then derive the pairwise trading amounts. Our PEM framework can privately perform all the computations among all the agents without a trusted third party. We prove the privacy, individual rationality, and incentive compatibility for the PEM framework. Finally, we conduct experiments on real datasets to validate the effectiveness and efficiency of the PEM.