The evolution of cooperation in nature and human societies depends crucially on how the benefits from cooperation are divided and whether individuals have complete information about their payoffs. We tackle these questions by adopting a methodology from economics called mechanism design. Focusing on reproductive skew as a case study, we show that full cooperation may not be achievable due to private information over individuals' outside options, regardless of the details of the specific biological or social interaction. Further, we consider how the structure of the interaction can evolve to promote the maximum amount of cooperation in the face of the informational constraints. Our results point to a distinct avenue for investigating how cooperation can evolve when the division of benefits is flexible and individuals have private information.