A promising feature of emerging wireless sensor networks is the opportunity for each spatially-distributed node to measure its local state and transmit only information relevant to effective global decision-making. An equally important design objective, as a result of each node's finite power, is for measurement processing to satisfy explicit constraints on, or perhaps make selective use of, the distributed algorithmic resources. We formulate this multi-objective design problem within the Bayesian decentralized detection paradigm, modeling resource constraints by a directed acyclic network with low-rate, unreliable communication links. Existing team theory establishes when necessary optimality conditions reduce to a convergent iterative algorithm to be executed offline (i.e., before measurements are processed). Even so, this offline algorithm has exponential complexity in the number of nodes, and its distributed implementation assumes a fully-connected communication network. We state conditions under which the offline algorithm admits an efficient message-passing interpretation, featuring linear complexity and a natural distributed implementation. We experiment with a simulated network of binary detectors, applying the message-passing algorithm to optimize the achievable tradeoff between global detection performance and network-wide online communication. The empirical analysis also exposes a design tradeoff between constraining in-network processing to preserve resources (per online measurement) and then having to consume resources (per offline reorganization) to maintain detection performance.