A key question in research on the neurobiology of language is to which extent the language production and comprehension systems share neural infrastructure, but this question has not been addressed in the context of conversation. We utilized a public fMRI dataset where 24 participants engaged in unscripted conversations with a confederate outside the scanner, via an audio-video link. We provide evidence indicating that the two systems share neural infrastructure in the left-lateralized perisylvian language network, but diverge regarding the level of activation in regions within the network. Activity in the left inferior frontal gyrus was stronger in production compared to comprehension, while comprehension showed stronger recruitment of the left anterior middle temporal gyrus and superior temporal sulcus, compared to production. Although our results are reminiscent of the classical Broca-Wernicke model, the anterior (rather than posterior) temporal activation is a notable difference from that model. This is one of the findings that may be a consequence of the conversational setting, another being that conversational production activated what we interpret as higher-level socio-pragmatic processes. In conclusion, we present evidence for partial overlap and functional asymmetry of the neural infrastructure of production and comprehension, in the above-mentioned frontal vs temporal regions during conversation.