Federated learning (FL) enables collaborative model training across decentralized clients without sharing local data, thereby enhancing privacy and facilitating collaboration among clients connected via social networks. However, these social connections introduce privacy externalities: a client's privacy loss depends not only on its privacy protection strategy but also on the privacy decisions of others, propagated through the network via multi-hop interactions. In this work, we propose a socially-aware privacy-preserving FL mechanism that systematically quantifies indirect privacy leakage through a multi-hop propagation model. We formulate the server-client interaction as a two-stage Stackelberg game, where the server, as the leader, optimizes incentive policies, and clients, as followers, strategically select their privacy budgets, which determine their privacy-preserving levels by controlling the magnitude of added noise. To mitigate information asymmetry in networked privacy estimation, we introduce a mean-field estimator to approximate the average external privacy risk. We theoretically prove the existence and convergence of the fixed point of the mean-field estimator and derive closed-form expressions for the Stackelberg Nash Equilibrium. Despite being designed from a client-centric incentive perspective, our mechanism achieves approximately-optimal social welfare, as revealed by Price of Anarchy (PoA) analysis. Experiments on diverse datasets demonstrate that our approach significantly improves client utilities and reduces server costs while maintaining model performance, outperforming both Social-Agnostic (SA) baselines and methods that account for social externalities.