Mobile edge computing (MEC) has empowered mobile devices (MDs) in supporting artificial intelligence (AI) applications through collaborative efforts with proximal MEC servers. Unfortunately, despite the great promise of device-edge cooperative AI inference, data privacy becomes an increasing concern. In this paper, we develop a privacy-aware multi-device cooperative edge inference system for classification tasks, which integrates a distributed bidding mechanism for the MEC server's computational resources. Intermediate feature compression is adopted as a principled approach to minimize data privacy leakage. To determine the bidding values and feature compression ratios in a distributed fashion, we formulate a decentralized partially observable Markov decision process (DEC-POMDP) model, for which, a multi-agent deep deterministic policy gradient (MADDPG)-based algorithm is developed. Simulation results demonstrate the effectiveness of the proposed algorithm in privacy-preserving cooperative edge inference. Specifically, given a sufficient level of data privacy protection, the proposed algorithm achieves 0.31-0.95% improvements in classification accuracy compared to the approach being agnostic to the wireless channel conditions. The performance is further enhanced by 1.54-1.67% by considering the difficulties of inference data.