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Federated learning (FL) allows multiple medical institutions to collaboratively learn a global model without centralizing client data. It is difficult, if possible at all, for such a global model to commonly achieve optimal performance for each individual client, due to the heterogeneity of medical images from various scanners and patient demographics. This problem becomes even more significant when deploying the global model to unseen clients outside the FL with unseen distributions not presented during federated training. To optimize the prediction accuracy of each individual client for medical imaging tasks, we propose a novel unified framework for both \textit{Inside and Outside model Personalization in FL} (IOP-FL). Our inside personalization uses a lightweight gradient-based approach that exploits the local adapted model for each client, by accumulating both the global gradients for common knowledge and the local gradients for client-specific optimization. Moreover, and importantly, the obtained local personalized models and the global model can form a diverse and informative routing space to personalize an adapted model for outside FL clients. Hence, we design a new test-time routing scheme using the consistency loss with a shape constraint to dynamically incorporate the models, given the distribution information conveyed by the test data. Our extensive experimental results on two medical image segmentation tasks present significant improvements over SOTA methods on both inside and outside personalization, demonstrating the potential of our IOP-FL scheme for clinical practice.