The Simon effect, that is the advantage of the spatial correspondence between stimulus and response locations when stimulus location is a task-irrelevant dimension, occurs even when the task is performed together by two participants, each performing a go/no-go task. Previous studies showed that this joint Simon effect, considered by some authors as a measure of self-other integration, does not emerge when during task performance co-actors are required to compete. The present study investigated whether and for how long competition experienced during joint performance of one task can affect performance in a following joint Simon task. In two experiments, we required pairs of participants to perform together a joint Simon task, before and after jointly performing together an unrelated non-spatial task (the Eriksen flanker task). In Experiment 1, participants always performed the joint Simon task under neutral instructions, before and after performing the joint flanker task in which they were explicitly required either to cooperate with (i.e., cooperative condition) or to compete against a co-actor (i.e., competitive condition). In Experiment 2, they were required to compete during the joint flanker task and to cooperate during the subsequent joint Simon task. Competition experienced in one task affected the way the subsequent joint task was performed, as revealed by the lack of the joint Simon effect, even though, during the Simon task participants were not required to compete (Experiment 1). However, prior competition no longer affected subsequent performance if a new goal that created positive interdependence between the two agents was introduced (Experiment 2). These results suggest that the emergence of the joint Simon effect is significantly influenced by how the goals of the co-acting individuals are related, with the effect of competition extending beyond the specific competitive setting and affecting subsequent interactions.