The amount of biomedical data continues to grow rapidly. However, the ability to collect data from multiple sites for joint analysis remains challenging due to security, privacy, and regulatory concerns. We present a Secure Federated Learning architecture, MetisFL, which enables distributed training of neural networks over multiple data sources without sharing data. Each site trains the neural network over its private data for some time, then shares the neural network parameters (i.e., weights, gradients) with a Federation Controller, which in turn aggregates the local models, sends the resulting community model back to each site, and the process repeats. Our architecture provides strong security and privacy. First, sample data never leaves a site. Second, neural parameters are encrypted before transmission and the community model is computed under fully-homomorphic encryption. Finally, we use information-theoretic methods to limit information leakage from the neural model to prevent a curious site from performing membership attacks. We demonstrate this architecture in neuroimaging. Specifically, we investigate training neural models to classify Alzheimer's disease, and estimate Brain Age, from magnetic resonance imaging datasets distributed across multiple sites, including heterogeneous environments where sites have different amounts of data, statistical distributions, and computational capabilities.