The Internet of Things (IoT) is gathering paces in the new era of Industry 4.0, and the Digital Twin (DT) technology bridges the gap between the bursting amounts of data generated by IoT devices and the user requirements for real-time data processing. DT services maintain living digital models of physical objects, and a DT network enables comprehensive service provisioning with the global knowledge of a group of DTs. On the other hand, exposing serverless computing at network edges, the recent advances in Mobile Edge Computing (MEC) introduce new inspirations to the DT landscape that ensure fine-grained resource management and low network-wide delay of DT services. However, social relationships among IoT devices and DT data privacy impact DT orchestrations. In this paper, we first design a differential privacy-based federated learning framework to build a DT network for DT services in response to user requests in an MEC, thereby enhancing the Quality of Services (QoS). Built upon the proposed framework, we then formulate two novel social-aware DT placement problems: the static social-aware S_DT placement problem, and the dynamic social-aware S_DT placement problem, respectively. We also show the NP-hardness of the defined problems. Then, we formulate an Integer Linear Program (ILP) solution to the static social-aware S_DT placement problem when the problem size is small; otherwise we develop an approximation algorithm with a provable approximation ratio for it. Third, we study the dynamic social-aware S_DT placement problem when requests arrive one by one without the knowledge of future request arrivals over the time horizon, for which we devise an online algorithm with a provable competitive ratio. Finally, we conduct simulations to evaluate the performance of the proposed algorithms. Simulation results show that the proposed algorithms outperform their counterparts, improving the performance compared with their baselines by no less than 14.9%.