Motivated by an example by Tenney and Sandell (1981), we discuss the trade-off between performance of local detectors (LDs) and the combined LD/Data Fusion Center system in parallel decision fusion architectures. In these architectures the LDs make observations, translate these observations to local decisions, and send these local decisions forward to a Data Fusion Center (DFC). The DFC uses the local decisions to synthesize a global decision (in our context both local and global decisions are binary and pertain to binary hypothesis testing based on the LD observations; in other words, both LDs and DFC decide whether to accept or reject a hypothesis). The original example demonstrated how the minimization of a global performance index by the combined system may yield an alignment of the local detectors that avoids a high value of the performance index, but otherwise have no value at the LD level (the LDs are directed to make constant decisions that are almost independent of the observations, in order to avoid a local-decision combination that would incur a high penalty). If we require that the global performance index be minimized while the LDs are also allowed to minimize a local performance index (or have constraints on their error probabilities), a trade-off emerges between the local and global performances. In this paper we provide an example similar in nature to the Tenney-Sandell example, and proceed to analyze the impact of performance constraints on the LDs on the design and performance of the parallel decision fusion architecture. If we provide reasonable constraints on the performance of the LDs, a compromise can be established between the global performance index and the local LD performances.