This technical note deals with the problem of designing a distributed fault detection methodology for distributed (and possibly large-scale) nonlinear dynamical systems that are modelled as the interconnection of several subsystems. The subsystems are allowed to overlap, thus sharing some state components. For each subsystem, a Local Fault Detector is designed, based on the measured local state of the subsystem as well as the transmitted variables of neighboring states that define the subsystem interconnections. The local detection decision is made on the basis of the knowledge of the local subsystem dynamic model and of an adaptive approximation of the interconnection with neighboring subsystems. The use of a specially-designed consensus-based estimator is proposed in order to improve the detectability of faults affecting variables shared among different subsystems. Simulation results provide an evidence of the effectiveness of the proposed distributed fault detection scheme.