We investigate welfare and equilibrium trading in a decentralized search market with asymmetric information and bilateral communication opportunities. Buyers and sellers meet randomly and pairwise and view a shared signal of the seller's quality. In the following signaling game, the sellers can thus either rely on this costless signal (pool) or costly signaling (separate). We characterize the full set of equilibria for exogenous market quality and outside options. We study what kinds of

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... can be sustained when average quality in the market and outside options arise endogenously in a stationary Markovian equilibrium. We show that in the lemons market (low entrant quality) only standard dynamics can arise (low quality is more liquid) but in a better market (high entrant quality) also inverse dynamics are possible (high quality is more liquid). We argue that both cases are relevant because neither market is efficient. JEL Classification: D82, D83