This paper concerns how trade mechanism choice affects how decentralized markets respond to shocks and policy choices. We consider this issue in the context of housing market search. We pose a dynamic search model in which agents can trade by auction or negotiation, both featuring two-sided incomplete information. We apply the model to housing data, estimating buyer and seller value distributions using a structural auction model, primitives that are used in solving for the search model equilibrium. Adding auctions as a second mechanism dampens the shock response of prices and values as agents optimally switch between mechanisms. We also find that policies that increase seller information disclosure at one mechanism can nonetheless benefit sellers and harm buyers, at odds with their intended purpose. Our estimates also highlight how mechanism efficiency assumptions influence search cost inference, with estimated seller negotiation search costs significantly lower under Nash bargaining than incomplete information. (With David Genesoveb and James Hansena)