We consider a Neyman-Pearson (NP) distributed binary detection problem in a bandwidth constrained wireless sensor network, where the fusion center (FC) is responsible for fusing signals received from sensors and making a final decision about the presence or absence of a signal source in correlated Gaussian noises. Given this signal model, our goals are (i) to investigate whether or not randomized transmission can improve detection performance, under communication rate constraint, and (ii) to explore how the correlation among observation noises would impact performance. To achieve these goals, we propose two novel schemes that combine the concepts of censoring and randomized transmission (which we name CRT-I and CRT-II schemes) and compare them with the pure censoring scheme. In CRT (pure censoring) schemes, we map randomly (deterministically) a sensor's observation to a ternary transmit symbol $u_k \in \{-1,0,1\}$ where ``$0$'' corresponds to no transmission (sensor censors). Assuming sensors transmit $u_k$'s over orthogonal fading channels, we formulate and address two system-level constrained optimization problems: in the first problem we minimize the probability of miss detection at the FC, subject to constraints on the probabilities of transmission and false alarm at the FC; in the second (dual) problem we minimize the probability of transmission, subject to constraints on the probabilities of miss detection and false alarm at the FC. Based on the expressions of the objective functions and the constraints in each problem, we propose different optimization techniques to address these two problems. Through analysis and simulations, we explore and provide the conditions (in terms of communication channel signal-to-noise ratio, degree of correlation among sensor observation noises, and maximum allowed false alarm probability) under which CRT schemes outperform pure censoring scheme.