The smart distribution grid (SDG), characterized by large-scale interconnections and strong dependence on information and communication technologies, is highly susceptible to potential security threats, such as spoofing attacks and man-in-the-middle attacks. These threats may lead to the leakage of sensitive user power-expenditure information, even cause great economic damage. Therefore, authentication is of utmost importance in guaranteeing the electrical safety of SDGs. In this paper, we present a distributed physical layer authentication (DPLA) scheme tailored for smart meter authentication. The scheme overcomes the limitations of traditional upper-layer cryptography-based mechanisms, and achieves lightweight continuous authentication in a cooperative manner. To fully exploit the channel information collected by collaborative nodes located in different azimuths, a CNN algorithm is designed for deep feature extraction. Moreover, a situational-aware dynamic weighted voting strategy is introduced to coordinate inconsistent opinions, thereby making unified decisions. Aimed at maximizing the integrated performance gains of DPLA, both long-term reputation and short-term performance are taken into account for node’s weight update. Finally, simulations are carried out. The results demonstrate that our scheme outperforms DPLAs based on static voting strategies with respect to authentication accuracy, anti-disturbance robustness and environmental adaptability; Hence, it caters to the demand for high-quality continuous authentication in SDGs.