Peer-to-peer (P2P) energy trading in active distribution networks (ADNs) plays a pivotal role in promoting the efficient consumption of renewable energy sources. However, it is challenging to effectively coordinate the power dispatch of ADNs and P2P energy trading while preserving the privacy of different physical interests. Hence, this paper proposes a soft actor-critic algorithm incorporating distributed trading control (SAC-DTC) to tackle the optimal power dispatch of ADNs and the P2P energy trading considering privacy preservation among prosumers. First, the soft actor-critic (SAC) algorithm is used to optimize the control strategy of device in ADNs to minimize the operation cost, and the primary environmental information of the ADN at this point is published to prosumers. Then, a distributed generalized fast dual ascent method is used to iterate the trading process of prosumers and maximize their revenues. Subsequently, the results of trading are encrypted based on the differential privacy technique and returned to the ADN. Finally, the social welfare value consisting of ADN operation cost and P2P market revenue is utilized as a reward value to update network parameters and control strategies of the deep reinforcement learning. Simulation results show that the proposed SAC-DTC algorithm reduces the ADN operation cost, boosts the P2P market revenue, maximizes the social welfare, and exhibits high computational accuracy, demonstrating its practical application to the operation of power systems and power markets.