The core use of human language is communicating complex ideas from one mind to another in everyday conversations. In conversations, comprehension and production processes are intertwined, as speakers soon become listeners, and listeners become speakers. Nonetheless, the neural systems underlying these faculties are typically studied in isolation using paradigms that cannot fully engage our capacity for interactive communication. Here, we used an fMRI hyperscanning paradigm to measure neural activity simultaneously in pairs of subjects engaged in real-time, interactive conversations. We used contextual word embeddings from a large language model to quantify the linguistic coupling between production and comprehension systems within and across individual brains. We found a highly overlapping network of regions involved in both production and comprehension spanning much of the cortical language network. Our findings reveal that shared representations for both processes extend beyond the language network into areas associated with social cognition. Together, these results suggest that the specialized neural systems for speech perception and production align on a common set of linguistic features encoded in a broad cortical network for language and communication.