Sharing synthetic medical images is a promising alternative to sharing real images that can improve patient privacy and data security. To get good results, existing methods for medical image synthesis must be manually adjusted when they are applied to unseen data. To remove this manual burden, we introduce a Hyperparameter-Free distributed learning method for automatic medical image Synthesis, Sharing, and Segmentation called HyFree-S3. For three diverse segmentation settings (pelvic MRIs, lung X-rays, polyp photos), the use of HyFree-S3 results in improved performance over training only with site-specific data (in the majority of cases). The hyperparameter-free nature of the method should make data synthesis and sharing easier, potentially leading to an increase in the quantity of available data and consequently the quality of the models trained that may ultimately be applied in the clinic.