Joint cognition refers to the mental systems that support group performance when carrying out a shared, or jointly owned task. We focused here on understanding the social configurations that underpin key phenomena in joint cognition, in particular, whether individual cognition in task-sharing environments is mostly shaped by social factors or not. To this end, we investigated, first and mainly, whether human presence is necessary for the creation of joint performance; second and separately, whether prior experience of task sharing has an adaptive influence on subsequent individual choices; and third and additionally, whether individual differences in a social trait mediate joint performance. We describe an experiment in which participants combined with another human or a computer as they attempted to generate a paired sequence that was as random as possible. First, we found little difference in joint performance with regard to whether a human or a computer was the co-participant, except for immediate repetitive response. Second, we found evidence for choice adaptation, but only under the lower time pressure. Third, we replicated previous research in which no systematic link was established between social desirability and joint performance. We conclude that joint cognition phenomena may be rooted primarily in turn-taking configurations rather than in social dynamics per se.