Effective social interactions rely on humans' ability to attune to others within social contexts. Recently, it has been proposed that the emergence of shared representations, as indexed by the Joint Simon effect (JSE), might result from interpersonal coordination (Malone et al., 2014). The present study aimed at examining interpersonal coordination in cooperative and competitive joint tasks. To this end, in two experiments we investigated response coordination, as reflected in instantaneous cross-correlation, when co-agents cooperate (Experiment 1) or compete against each other (Experiment 2). In both experiments, participants performed a go/no-go Simon task alone and together with another agent in two consecutive sessions. In line with previous studies, we found that social presence differently affected the JSE under cooperative and competitive instructions. Similarly, cooperation and competition were reflected in co-agents response coordination. For the cooperative session (Experiment 1), results showed higher percentage of interpersonal coordination for the joint condition, relative to when participants performed the task alone. No difference in the coordination of responses occurred between the individual and the joint conditions when co-agents were in competition (Experiment 2). Finally, results showed that interpersonal coordination between co-agents implies the emergence of the JSE. Taken together, our results suggest that shared representations seem to be a necessary, but not sufficient, condition for interpersonal coordination.