Recent booming development of Generative Artificial Intelligence (GenAI) has facilitated an emerging model commercialization for the purpose of reinforcement on model performance, such as licensing or trading Deep Neural Network (DNN) models. However, DNN model trading may trigger concerns of the unauthorized replications or misuses over the model, so that the benefit of the model ownership will be violated. Model identity auditing is a challenging issue in protecting intellectual property of DNN models and verifying the integrity and ownership of models for guaranteeing trusts in transactions is one of the critical obstacles. In this paper, we focus on the above issue and propose a novel Accumulator-enabled Auditing for Distributed Identity of DNN Model (A2-DIDM) that utilizes blockchain and zero-knowledge techniques to protect data and function privacy while ensuring the lightweight on-chain ownership verification. The proposed model presents a scheme of identity records via configuring model weight checkpoints with corresponding zero-knowledge proofs, which incorporates predicates to capture incremental state changes in model weight checkpoints. Our scheme ensures both computational integrity of DNN training process and programmability, so that the uniqueness of the weight checkpoint sequence in a DNN model is preserved, ensuring the correctness of the model identity auditing. In addition, A2-DIDM also addresses privacy protections in distributed identity via a proposed method of accumulators. We systematically analyze the security and robustness of our proposed model and further evaluate the effectiveness and usability of auditing DNN model identities.