We propose and investigate a semantics for peer data exchange systems where different peers are related by data exchange constraints and trust relationships. These two elements plus the data at the peers' sites and their local integrity constraints are made compatible via a semantics that characterizes sets of solution instances for the peers. They are the intended – possibly virtual – instances for a peer that are obtained through a data repair semantics that we introduce and investigate. The semantically correct answers from a peer to a query, the so-called peer consistent answers, are defined as those answers that are invariant under all its different solution instances. We show that solution instances can be specified as the models of logic programs with a stable model semantics. The repair semantics is based on null values as used in SQL databases, and is also of independent interest for repairs of single databases with respect to integrity constraints.