We study games in which several principals contract with several privately-informed agents. We show that enabling the principals to engage in contractible private disclosures by sending private signals to the agents about how the mechanisms will respond to the agents' messages-can significantly affect the predictions of such games. Our first result shows that private disclosures may generate equilibrium outcomes that cannot be supported in any game without private disclosures, no matter the richness of the message spaces and the availability of public randomizing devices. The result thus challenges the canonicity of the universal mechanisms of Epstein and Peters (1999). Our second result shows that equilibrium outcomes of games without private disclosures need not be sustainable when private disclosures are allowed. The result thus challenges the robustness of the "folk theorems" of Yamashita (2010) and Peters and Troncoso-Valverde (2013). These findings call for a novel approach to the analysis of competing-mechanism games.