Existing channel-aware decision fusion schemes assume that the local detection probability is known at the fusion center (FC). However, this paradigm ignores the possibility of unknown sensor alarm responses to the occurrence of events. Accordingly, this correspondence examines the binary decision fusion problem under the assumption that the local detection probability is unknown. Treating the communication links between the nodes and the FC as binary symmetric channels and assuming that the sensor nodes transmit simple one-bit reports to the FC, the global fusion rule is formulated initially in terms of the generalized likelihood ratio test (GLRT). Adopting the assumption of a high SNR regime, an approximate maximum likelihood (ML) estimate is derived for the unknown parameter required to implement the GLRT that is affine in the received data. The GLRT-based formulation is intuitively straightforward, but does not permit a tractable performance analysis. Therefore, motivated by the affine nature of the approximate ML solution, a simple alternative fusion rule is proposed in which the test statistic remains affine in the received data. It is shown that the proposed fusion rule facilitates the analytic characterization of the channel effect on the global detection performance. In addition, given a reasonable range of the local detection probability, it is shown that the global detection probability can be improved by reducing the total link error. Thus, a sensor power allocation scheme is proposed for enhancing the detection performance by improving the link quality. Simulation results show that: 1) the alternative fusion rule outperforms the GLRT; and 2) the detection performance of the fusion rule is further improved when the proposed power loading method is applied.