Distributed systems are fundamental to large-scale personalized recommendation services, enabling collaborative learning across decentralized data sources. Federated recommendation, as a distributed paradigm, leverages Graph Neural Networks (GNNs) to capture high-order interactions while preserving user data locality. However, ensuring user privacy in such distributed environments remains a significant challenge. Existing solutions typically employ differential privacy by injecting fixed noise into client gradients, but this approach often struggles to balance privacy protection and model utility—leading to either degraded recommendation accuracy or insufficient privacy guarantees. To address these limitations, we propose Adap DP-FR, a distributed federated GNN-based recommendation framework that jointly optimizes model performance and privacy. Central to our framework is an Adaptive Sensitivity-based Differential Privacy (ASDP) mechanism, which dynamically assesses the sensitivity of client gradients, allocates privacy budgets accordingly, and injects noise adaptively. This design effectively minimizes utility loss while providing strong privacy guarantees. Extensive experiments on two real-world recommendation datasets demonstrate that, under the same privacy budget, Adap DP-FR reduces RMSE by 2% and lowers the membership inference attack success rate by approximately 67%, significantly outperforming existing dynamic budget allocation baselines.