This paper considers the problem of hypothesis testing through the distributed sampling of a remote Poisson source. More specifically, we consider the scenario in which one of two Poisson sources is observed at a set of K remote observers. These source observations are subject to erasure-type noise, so some of the source spikes are not received at some of the K observers, leading to incomplete signal reception. The partially received signal is then transmitted to a central detector whose task is to identify the originating source. A crucial constraint in our study is the limited capacity for signal forwarding from the observers to the central detector. We assume that these remote observers are subject to a sampling constraint so that only a portion of the total signal received at all remote observers can be forwarded to the central detector. Given this sampling constraint, we determine the optimal sampling strategy at the remote observers that minimizes the probability of error in the detection of the remote source. This problem setting is motivated by the problem of testing of large populations through multiple tests, each subject to a certain false positive and false negative rates. Our paper contributes to the field through a comprehensive mathematical analysis, providing innovative strategies and insights for efficient resource allocation in large-scale testing. The proposed model not only enhances understanding of distributed Poisson sampling under constraints but also offers practical applications in robust decision-making for hypothesis testing in complex environments.