Sleep staging is essential in assessing sleep quality and diagnosing sleep-related disorders, but the lack of labeled data impedes the development of automatic sleep staging models. Generally, institutions rely on semi-supervised approaches to enhance the utilization of their own unlabeled data. However, the task knowledge obtained from a limited amount of labeled data is often insufficient to guide the learning based on large amounts of unlabeled data, which may even lead to catastrophic forgetting and further degrade the performance of most existing methods. In this paper, we propose a novel strategy of building secure collaboration among multiple institutions, to achieve the implicit augmentation of labeled data and expansion of task knowledge for each participating institution by acquiring external knowledge from others. We adopt the Federated Learning (FL) to facilitate secure collaboration and propose a federated semi-supervised sleep staging method based on knowledge sharing, which enables the automatic scoring of sleep stages using only single-channel EEG data. The task knowledge in our method is contained in relationships, which exist naturally among sleep stages and can be extracted from both local labeled and unlabeled data. Furthermore, the knowledge sharing among participating institutions can be achieved by aligning the local relationships to the aggregated global relationships. Additionally, we employ prototype-contrastive learning to enhance the clarity of relationships extracted from labeled data, and propose pseudo-labeling optimization to generate reliable pseudo-labels for subsequent relationship extraction from unlabeled data. Our method is shown to be effective and outperforms compared methods in extensive experiments conducted on two publicly available datasets.