Antibodies are proteins in the immune system which bind to antigens to mark them out for destruction. The binding sites in an antibody-antigen interaction are known as the paratope and epitope, respectively, and the prediction of these regions is key to vaccine and synthetic antibody development. Contrary to prior art, we argue that paratope and epitope predictors require asymmetric treatment, and propose distinct neural message passing architectures that are geared towards the specific aspects of paratope and epitope prediction, respectively. We obtain significant improvements on both tasks, setting the new state-of-the-art and recovering favourable qualitative predictions on antigens of relevance to COVID-19.