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Data analysis and monitoring on smart grids are jeopardized by attacks on cyber-physical systems. False data injection attack (FDIA) is one of the classes of those attacks that target the smart measurement devices by injecting malicious data. The employment of machine learning techniques in the detection and localization of FDIA is proven to provide effective results. Training of such models requires centralized processing of sensitive user data that may not be plausible in a practical scenario. By employing federated learning for the detection of FDIA attacks, it is possible to train a model for the detection and localization of the attacks while preserving the privacy of sensitive user data. However, federated learning introduces new problems such as the personalization of the detectors in each node. In this paper, we propose a federated learning-based scheme combined with a hybrid deep neural network architecture that exploits the local correlations between the connected power buses by employing graph neural networks as well as the temporal patterns in the data by using LSTM layers. The proposed mechanism offers flexible and efficient training of an FDIA detector in a distributed setup while preserving the privacy of the clients. We validate the proposed architecture by extensive simulations on the IEEE 57, 118, and 300 bus systems and real electricity load data.