Epidemic prediction is of practical significance in public health, enabling early intervention, resource allocation, and strategic planning. However, privacy concerns often hinder the sharing of health data among institutions, limiting the development of accurate prediction models. In this paper, we develop a general privacy-preserving framework for node-level epidemic prediction on networks based on federated learning (FL). We frame the spatio-temporal spread of epidemics across multiple data-isolated subnetworks, where each node state represents the aggregate epidemic severity within a community. Then, both the pure temporal LSTM model and the spatio-temporal model i.e., Spatio-Temporal Graph Attention Network (STGAT) are proposed to address the federated epidemic prediction. Extensive experiments are conducted on various epidemic processes using a practical airline network, offering a comprehensive assessment of FL efficacy under diverse scenarios. By introducing the efficacy energy metric to measure system robustness under various client configurations, we systematically explore key factors influencing FL performance, including client numbers, aggregation strategies, graph partitioning, missing infectious reports. Numerical results manifest that STGAT excels in capturing spatio-temporal dependencies in dynamic processes whereas LSTM performs well in simpler pattern. Moreover, our findings highlight the importance of balancing feature consistency and volume uniformity among clients, as well as the prediction dilemma between information richness and intrinsic stochasticity of dynamic processes. This study offers practical insights into the efficacy of FL scenario in epidemic management, demonstrates the potential of FL to address broader collective dynamics.