A recent paper by Pozdnyakov and Steele [6] is devoted to distributed sensor networks in which the sensors send summarized information to some remotely located fusion agents. Main focus is on sensors that send their information according to a so-called binary-plus-passive design. Some exact relations in a sequential approach, aimed at making decisions before a given budget is exhausted are presented. In this paper, we provide asymptotic results in the case of more general sensors for increasing budgets. The essential ingredient is the observation that the setting can be modeled within the framework of certain stopped two-dimensional random walks. We also provide asymptotics for the cost for large decision boundaries, and, finally, some comments on the probability of a positive/negative decision as well as on the duration of the process until a decision is made in the binary-plus-passive design.