Human communication has been described as a contextual social inference process. Research into great ape communication has been inspired by this view to look for the evolutionary roots of the social, cognitive and interactional processes involved in human communication. This approach has been highly productive, yet it is partly compromised by the widespread focus on how great apes use and understand individual signals. This paper introduces a computational model that formalizes great ape communication as a multi-faceted social inference process that integrates (a) information contained in the signals that make up an utterance, (b) the relationship between communicative partners and (c) the social context. This model makes accurate qualitative and quantitative predictions about real-world communicative interactions between semi-wild-living chimpanzees. When enriched with a pragmatic reasoning process, the model explains repeatedly reported differences between humans and great apes in the interpretation of ambiguous signals (e.g. pointing or iconic gestures). This approach has direct implications for observational and experimental studies of great ape communication and provides a new tool for theorizing about the evolution of uniquely human communication. This article is part of the theme issue 'Revisiting the human 'interaction engine': comparative approaches to social action coordination'.