This article analyses the influence of trust on the functioning of a market for perishable goods, where there exists no quality signal and quantities can be scarce. On this market, agents can choose between bidding or exchanging through bilateral transactions. It is well accepted in economy that trust plays an important role in transactions but its definition and measurement stay, as far as we know, very elusive. Starting from the empirical analysis of a market with a peculiar organization, the Boulogne-sur-mer fish market, where people have the choice between trading through auctions or bilateral exchanges, we propose a measurement of trust, based on the dynamics of agents' encounters. We then analyze the differences in the social network structures and estimate how they affects the market outcomes. We bring into the light that, when the transaction links on the auction market reflects the economic constraints of the partners, the relationships on the bilateral market depends on something more. Clearly, the prices of the bilateral transactions are the consequences of economics and non economics determinants. At first glance, the stable coexistence of two market structures looks like a paradox. Our results help to understand the distinctive characteristics and functioning of each sub-market. This discussion contributes to the debate about the efficiency of market structures.