They study insurance markets with nonexclusive contracts, introducing bilateral endogenous information disclosure about insurance sales and purchases by firms and consumers. They show that a competitive equilibrium exists under remarkably mild conditions, and characterize the unique equilibrium outcome. With two types of consumers the outcome consists of a pooling contract which maximizes the well-being of the low risk type (along the zero profit pooling line) plus a supplemental (undisclosed and nonexclusive) contract that brings the high risk type to full insurance (at his own odds). They show that this outcome is extremely robust and constrained Pareto efficient. Consumer disclosure and asymmetric equilibrium information flows are critical in supporting the equilibrium.