Cooperation and competition are two ways of social interaction keys to life in society. Recent EEG-based hyperscanning studies reveal that cooperative and competitive interactions induce an increase in interbrain coupling. However, whether this interbrain coupling effect is just a reflection of inter-subject motor coordination or can also signal the type of social interaction is unknown. Here, we show that behavioral coordination and social interaction type can be distinguished according to the frequency of oscillation in which the brains are coupled. We use EEG to simultaneously measure the brain activity of pairs of subjects, while they were performing a visual cue-target task in a cooperative and competitive manner. Behavioral responses were quasi-simultaneous between subject pairs for both competitive and cooperative conditions, with faster average response times for the competitive condition. Concerning brain activity, we found increased interbrain coupling in theta band (3–7 Hz) during cooperation and competition, with stronger coupling during competitive interactions. This increase of interbrain theta coupling correlated with a decrease in reaction times of the dyads. Interestingly, we also found an increase in brain-to-brain coupling in gamma band (38–42 Hz) only during cooperative interactions. Unlike the theta coupling effect, the gamma interbrain coupling did not correlate with dyads’ reaction times. Taken together, these results suggest that theta interbrain coupling could be linked to motor coordination processes common to cooperative and competitive interactions, while gamma brain-to-brain coupling emerges as an electrophysiological marker of shared intentionality during cooperative interactions.