Multi-institutional efforts can facilitate training of deep MRI reconstruction models, albeit privacy risks arise during cross-site sharing of imaging data. Federated learning (FL) has recently been introduced to address privacy concerns by enabling distributed training without transfer of imaging data. Existing FL methods for MRI reconstruction employ conditional models to map from undersampled to fully-sampled acquisitions via explicit knowledge of the imaging operator. Since conditional models generalize poorly across different acceleration rates or sampling densities, imaging operators must be fixed between training and testing, and they are typically matched across sites. To improve generalization and flexibility in multi-institutional collaborations, here we introduce a novel method for MRI reconstruction based on Federated learning of Generative IMage Priors (FedGIMP). FedGIMP leverages a two-stage approach: cross-site learning of a generative MRI prior, and subject-specific injection of the imaging operator. The global MRI prior is learned via an unconditional adversarial model that synthesizes high-quality MR images based on latent variables. Specificity in the prior is preserved via a mapper subnetwork that produces site-specific latents. During inference, the prior is combined with subject-specific imaging operators to enable reconstruction, and further adapted to individual test samples by minimizing data-consistency loss. Comprehensive experiments on multi-institutional datasets clearly demonstrate enhanced generalization performance of FedGIMP against site-specific and federated methods based on conditional models, as well as traditional reconstruction methods.