The limited availability of sleep data collected by individual institutions poses a significant challenge in developing automatic sleep staging models, especially given that most existing models rely heavily on data-driven approaches. The advent of federated learning has introduced an innovative and reliable paradigm for inter-institutional collaboration, facilitating task knowledge sharing and implicit data augmentation while preserving privacy. However, in traditional federated learning methodologies, local models are typically initialized with the global model at the onset of each iteration. This global-knowledge-centric approach often results in catastrophic forgetting during local training and inadequately adapts to the diversity of local data, compromising the overall generalization performance of the model. In this paper, we introduce a novel re-aggregation strategy for local model initialization that synergizes the global model with historical local models, thereby mitigating the undue influence of global knowledge and preserving local task-specific information. The re-aggregation weights are adaptively determined based on the confidence of the global and historical models on local datasets. Furthermore, to address the inherent ambiguity among sleep stages, a dual prototype-contrastive learning module is proposed, comprising Prototype-Consistency Contrastive (PCC) and Prototype-Distinguishability Contrastive (PDC) components. Specifically, the PCC component is designed to ensure consistency between local prototypes and unbiased prototypes derived from re-aggregation, enhancing intra-class knowledge coherence. The goal of the PDC component is to strengthen the discriminability among various sleep stages, improving inter-class differentiation. Comprehensive experiments conducted on two public datasets demonstrate the effectiveness, superiority, and flexibility of the proposed method.