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Supervised deep learning needs a large amount of labeled data to achieve high performance. However, in medical imaging analysis, each site may only have a limited amount of data and labels, which makes learning ineffective. Federated learning (FL) can learn a shared model from decentralized data. But traditional FL requires fully-labeled data for training, which is very expensive to obtain. Self-supervised contrastive learning (CL) can learn from unlabeled data for pre-training, followed by fine-tuning with limited annotations. However, when adopting CL in FL, the limited data diversity on each site makes federated contrastive learning (FCL) ineffective. In this work, we propose two federated self-supervised learning frameworks for volumetric medical image segmentation with limited annotations. The first one features high accuracy and fits high-performance servers with high-speed connections. The second one features lower communication costs, suitable for mobile devices. In the first framework, features are exchanged during FCL to provide diverse contrastive data to each site for effective local CL while keeping raw data private. Global structural matching aligns local and remote features for a unified feature space among different sites. In the second framework, to reduce the communication cost for feature exchanging, we propose an optimized method FCLOpt that does not rely on negative samples. To reduce the communications of model download, we propose the predictive target network update (PTNU) that predicts the parameters of the target network. Based on PTNU, we propose the distance prediction (DP) to remove most of the uploads of the target network. Experiments on a cardiac MRI dataset show the proposed two frameworks substantially improve the segmentation and generalization performance compared with state-of-the-art techniques.