We experimentally study decentralized one-to-one matching markets with transfers. We vary the information available to participants, complete or incomplete, and the surplus structure, supermodular or submodular. Several insights emerge. First, while markets often culminate in efficient matchings, stability is more elusive, reflecting the difficulty of arranging attendant transfers. Second, incomplete information and submodularity present hurdles to efficiency and especially stability, their combination drastically diminishes stability’s likelihood. Third, matchings form “from the top down” in complete-information supermodular markets, but exhibit many more and less-obviously ordered offers otherwise. Last, participants’ market positions matter far more than their dynamic bargaining styles for outcomes.