Computational prediction of protein structure from amino acid sequence alone has been achieved with unprecedented accuracy, yet the prediction of protein-protein interactions remains a challenge. Here, we assess the ability of protein language models (PLMs), routinely applied to protein folding, to be retrained for protein-protein interaction prediction. Existing models that exploit PLMs use a pre-trained PLM feature set, ignoring that the proteins are physically interacting. We propose PLM-interact, which goes beyond single proteins by jointly encoding protein pairs to learn their relationships, analogous to the next-sentence prediction task from natural language processing. This approach achieves state-of-the-art performance in a widely adopted cross-species protein-protein interaction prediction benchmark: trained on human data and tested on mouse, fly, worm, E. coli and yeast. In addition, we develop a fine-tuning method for PLM-interact to detect mutation effects on interactions. Finally, we report that the model outperforms existing approaches in predicting virus-host interaction at the protein level. Our work demonstrates that large language models can be extended to learn the intricate relationships among biomolecules from their sequences alone.