Smart meters play a crucial role in accurately measuring power usage and distribution data in microgrids. However, fine-grained data collection raises significant privacy concerns. To address this, it is essential to develop a distributed power trading system that is both trustworthy and privacy-preserving. In this paper, we propose a peer-to-peer power trading mechanism that ensures privacy through identity verification based on decentralized identifiers, zero-knowledge proofs, and pseudonyms. Besides, we introduce an efficient claims verification scheme with Boneh-Lynn-Shacham aggregate signature. We also validate the security of our proposed mechanism through the Real-or-Random model.