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Medical image segmentation under federated learning (FL) is a promising direction by allowing multiple clinical sites to collaboratively learn a global model without centralizing datasets. However, using a single model to adapt to various data distributions from different sites is extremely challenging. Personalized FL tackles this issue by only utilizing partial model parameters shared from global server, while keeping the rest to adapt to its own data distribution in the local training of each site. However, most existing methods concentrate on the partial parameter splitting, while do not consider the \textit{inter-site in-consistencies} during the local training, which in fact can facilitate the knowledge communication over sites to benefit the model learning for improving the local accuracy. In this paper, we propose a personalized federated framework with \textbf{L}ocal \textbf{C}alibration (LC-Fed), to leverage the inter-site in-consistencies in both \textit{feature- and prediction- levels} to boost the segmentation. Concretely, as each local site has its alternative attention on the various features, we first design the contrastive site embedding coupled with channel selection operation to calibrate the encoded features. Moreover, we propose to exploit the knowledge of prediction-level in-consistency to guide the personalized modeling on the ambiguous regions, e.g., anatomical boundaries. It is achieved by computing a disagreement-aware map to calibrate the prediction. Effectiveness of our method has been verified on three medical image segmentation tasks with different modalities, where our method consistently shows superior performance to the state-of-the-art personalized FL methods. Code is available at https://github.com/jcwang123/FedLC.