The paper proposes a logic framework for modeling the interaction among deductive databases and computing consistent answers to logic queries in a P2P environment. As usual, data are exchanged among peers by using logical rules, called mapping rules. The novelty of our approach is that only data not violating integrity constraints are exchanged. The (declarative) semantics of a P2P system is defined in terms of weak models. Under this semantics only facts not making the local databases inconsistent can be imported, and the preferred weak models are the consistent scenarios in which peers import maximal sets of facts not violating integrity constraints. A characterization of the preferred weak model semantics, allowing to model a P2P system with a prioritized logic program, is provided. The proposed framework is then extended in order to also take into account P2P system in which each peer may be locally inconsistent, i.e. its data does not satisfy some of its constraints. Finally, the complexity of P2P logic queries is investigated.