With increasing prosumers employed with distributed energy resources (DER), advanced energy management has become increasingly important. To this end, integrating demand-side DER into electricity market is a trend for future smart grids. The double-side auction (DA) market is viewed as a promising peer-to-peer (P2P) energy trading mechanism that enables interactions among prosumers in a distributed manner. To achieve the maximum profit in a dynamic electricity market, prosumers act as price makers to simultaneously optimize their operations and trading strategies. However, the traditional DA market is difficult to be explicitly modelled due to its complex clearing algorithm and the stochastic bidding behaviors of the participants. For this reason, in this paper we model this task as a multi-agent reinforcement learning (MARL) problem and propose an algorithm called DA-MADDPG that is modified based on MADDPG by abstracting the other agents’ observations and actions through the DA market public information for each agent’s critic. The experiments show that 1) prosumers obtain more economic benefits in P2P energy trading w.r.t. the conventional electricity market independently trading with the utility company; and 2) DA-MADDPG performs better than the traditional Zero Intelligence (ZI) strategy and the other MARL algorithms, e.g., IQL, IDDPG, IPPO and MADDPG.