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Medical image segmentation is challenging because of lack of annotated data and the reluctance of institutions sharing sensitive patient information. Recent advances in federated learning and semi-supervised learning enable training models with limited labels in a privacy-preserving manner. We propose FedWeP, a method for federated semi-supervised segmentation. FedWeP uses Randomized Weight Perturbation in which the server modulates model weights with Gaussian noise, and disseminates perturbed models to clients for semi-supervised training. Moreover, in order to facilitate consistency regularization at clients, models are perturbed in a controlled way so that the predictions from perturbed models do not deviate excessively, for which we propose similarity-based seed search. Importantly, our method requires only a single model to be exchanged between server and clients, unlike conventional semi-supervised methods using two models, halving the communication costs. We validate our approach using the COVID-19 infection dataset, and demonstrate that FedWeP outperforms the baseline using one model, and is comparable to those using two models, striking a balance between segmentation performance and communication costs.