In this article, we consider the detection of a localized source emitting a signal using a wireless sensor network (WSN). We consider that geographically distributed sensor nodes obtain energy measurements and compute cooperatively and in a distributed manner a statistic to decide if the source is present or absent without the need of a central node or fusion center (FC). We first start from the continuous-time signal sensed by the nodes and obtain an equivalent discrete-time hypothesis testing problem. Secondly, we propose a fully distributed scheme, based on the well-known generalized likelihood ratio (GLR) test, which is suitable for a WSN, where resources such as energy and communication bandwidth are typically scarce. In third place, we consider the asymptotic performance of the proposed GLR test. The derived results provide an excellent matching with the scenario in which only a finite amount of measurements are available at each sensor node. We finally show that the proposed distributed algorithm performs as well as the global GLR test in the considered scenarios, requiring only a small number of communication exchanges between nodes and a limited knowledge about the network structure and its connectivity.