We consider distributed detection with a large number of identical binary sensors deployed over a region where the phenomenon of interest (POI) has spatially varying signal strength. Each sensor makes a binary decision based on its own measurement, and the local decision of each sensor is sent to a fusion center using a random access protocol. The fusion center decides whether the event has occurred under a global size constraint in the Neyman-Pearson formulation. Assuming homogeneous Poisson distributed sensors, we show that the distribution of "alarmed" sensors satisfies the local asymptotic normality (LAN). We then derive an asymptotically locally most powerful (ALMP) detector optimized jointly over the fusion form and the local sensor threshold under the Poisson regime. We establish conditions on the spatial signal shape that ensure the existence of the ALMP detector. We show that the ALMP test statistic is a weighted sum of local decisions, the optimal weights being the shape of the spatial signal; the exact value of the signal strength is not required. We also derive the optimal threshold for each sensor. For the case of independent, identically distributed (iid) sensor observations, we show that the counting-based detector is also ALMP under the Poisson regime. The performance of the proposed detector is evaluated through analytic results and Monte Carlo simulations and compared with that of the counting-based detector. The effect of mismatched signal shapes is also investigated.