In this article, we consider the problem of distributed detection of a localized radio source emitting a signal. We consider that geographically distributed sensor nodes obtain energy measurements and compute cooperatively a statistic to decide if the source is present or absent. We model the radio source as a stochastic signal and deal with spatially statistically dependent measurements, whose probability density function (PDF) has unknown positive parameters when the radio source is active. Under the framework of the Generalized Likelihood Ratio Test (GLRT) theory, the positive constraint on the unknown multidimensional parameter makes the computation of the GLRT asymptotic performance (when the amount of sensor measurements tends to infinity) more involved. Nevertheless, we analytically characterize the asymptotic distribution of the statistic. Moreover, as the GLRT is not amenable for distributed settings because of the spatial statistical dependence of the measurements, we study a GLRT-like test where the joint PDF of the measurements is substituted by the product of its marginal PDFs, and therefore, the statistical dependence is completely discarded for building this test. Nevertheless, its asymptotic performance is proved to be identical to the original GLRT, showing that the statistically dependence of the measurements has no impact on the detection performance in the asymptotic scenario. Furthermore, the GLRT-like algorithm has a low computational complexity and demands low communication resources, as compared to the GLRT.