By analyzing protein-protein interaction (PPI) networks, one can find that a protein may have multiple binding partners. However, it is difficult to determine whether the interactions with these partners occur simultaneously from binary PPIs alone. Here, we construct the yeast and human competition-cooperation relationship networks (CCRNs) based on protein structural interactomes to clearly exhibit the relationship (competition or cooperation) between two partners of the same protein. If two partners compete for the same interaction interface, they would be connected by a competitive edge; otherwise, they would be connected by a cooperative edge. The properties of three kinds of hubs (i.e., competitive, modest, and cooperative hubs) are analyzed in the CCRNs. Our results show that competitive hubs have higher clustering coefficients and form clusters in the human CCRN, but these tendencies are not observed in the yeast CCRN. We find that the human-specific proteins contribute significantly to these differences. Subsequently, we conduct a series of computational experiments to investigate the regulatory mechanisms that avoid competition between proteins. Our comprehensive analyses reveal that for most yeast and human protein competitors, transcriptional regulation plays an important role. Moreover, the human-specific proteins have a particular preference for other regulatory mechanisms, such as alternative splicing.