This paper introduces a novel privacy-enhanced over-the-air Federated Learning (OTA-FL) framework using client-driven power balancing (CDPB) to address privacy concerns in OTA-FL systems. In recent studies, a server determines the power balancing based on the continuous transmission of channel state information (CSI) from each client. Furthermore, they concentrate on fulfilling privacy requirements in every global iteration, which can heighten the risk of privacy exposure as the learning process extends. To mitigate these risks, we propose two CDPB strategies -- CDPB-n (noisy) and CDPB-i (idle) -- allowing clients to adjust transmission power independently, without sharing CSI. CDPB-n transmits noise during poor conditions, while CDPB-i pauses transmission until conditions improve. To further enhance privacy and learning efficiency, we show a mixed strategy, CDPB-mixed, which combines CDPB-n and CDPB-i. Our experimental results show that CDPB outperforms traditional approaches in terms of model accuracy and privacy guarantees, providing a practical solution for enhancing OTA-FL in resource-constrained environments.