This study aimed to deepen our understanding of the behavioral and neurocognitive processes involved in human-human and human-robot communication in a more ecologically valid setting compared to the traditional neurolinguistic paradigms. We collected a novel open-source dataset (N=30 for human-human and N=20 for human-robot interactions), that includes fMRI, eye-tracking, segmented audio, video, and behavioral data, resulting in 30 minutes of free conversations per participant. To enable unrestricted, spontaneous robot behavior, we employed a novel VR-mediated teleoperation system. Our mixed design allowed us to compare participants' perception of humans and robots across three within-subject conditions of conversational engagement: Engaged Communicator, Active Listener, and Passive Listener. We provide an open-access dataset, replicable code for the teleoperation system, and an initial analysis of fMRI, behavioral, and speech data. We observed distinct neural profiles: speaking to the human agent recruited more higher-level frontal regions associated with socio-pragmatic processes, while listening to the robot recruited more sensory areas, including auditory and visual regions. Engagement levels and agent types also affected speech and behavioral patterns, offering valuable insights into conversational dynamics in human-human and human-robot interactions.