We study the problem of a seller (e.g., a bank) who is privately informed about the quality of her asset and wants to exploit gains from trade with uninformed buyers (e.g., investors) by issuing securities backed by her asset cash flows. In our setting, buyers post menus of contracts to screen the seller, but the seller cannot commit to trade with only one buyer, i.e., markets are non-exclusive. We show that non-exclusive markets behave very differently from exclusive ones: (i) separating contracts are never part of equilibrium; (ii) mispricing of claims faced by the seller is always greater than in exclusive markets; (iii) there is always a semi-pooling equilibrium where all sellers issue the same debt contract priced at average-valuation, and sellers of low-quality assets issue remaining cash flows at low-valuation; (iv) market liquidity can be higher or lower than in exclusive markets, but (v) the average quality of originated assets is always lower. Our model’s predictions are consistent with empirical evidence on issuance and pricing of mortgage-backed securities, and we use the theory to evaluate recent reforms aimed at enhancing transparency and exclusivity in markets.