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The existence of completely aligned and paired multi-modal neuroimaging data has proved its effectiveness in the diagnosis of brain diseases. However, collecting the full set of well-aligned and paired data is impractical, since the practical difficulties may include high cost, long time acquisition, image corruption, and privacy issues. Previously, the misaligned unpaired neuroimaging data (termed as MUD) are generally treated as noisy label. However, such a noisy label-based method fail to accomplish well when misaligned data occurs distortions severely. For example, the angle of rotation is different. In this paper, we propose a novel federated self-supervised learning (FedMed) for brain image synthesis. An affine transform loss (ATL) was formulated to make use of severely distorted images without violating privacy legislation for the hospital. We then introduce a new data augmentation procedure for self-supervised training and fed it into three auxiliary heads, namely auxiliary rotation, auxiliary translation and auxiliary scaling heads. The proposed method demonstrates the advanced performance in both the quality of our synthesized results under a severely misaligned and unpaired data setting, and better stability than other GAN-based algorithms. The proposed method also reduces the demand for deformable registration while encouraging to leverage the misaligned and unpaired data. Experimental results verify the outstanding performance of our learning paradigm compared to other state-of-the-art approaches.