We address two problems of distributed detection of a weak signal from dependent observations. In the first problem, two detectors must decide on the basis of their observations whether a weak signal is present or not. The observations of the two detectors consist of a common weak signal disturbed by two independent additive m-dependent or hi-mixing noise processes. Fixed-sample- size (block) detection is employed. The decisions are coupled through a common cost function, which consists of the sum of the error probabilities under the two hypotheses. In the second problem, the observations of each individual detector still consist of a common weak signal disturbed by an additive m- dependent or hi-mixing noise process, but the noise processes of the two detectors are now correlated. The cost function has a structure similar to that of the first problem. In both cases, the detectors employ suboptimal decision tests based on memoryless nonlinearities. Since the signal is weak, large sample sizes are necessary to guarantee high quality tests and the asymptotic performance is of interest. To determine the optimal nonlinearities for the two detectors, we identify new performance measures based on twodimensional Chernoff bounds, which correspond to the asymptotic relative efficiency (ARE) used for single-detector problems, and whose maximization implies the minimization of the aforementioned average cost function. This optimization results in integral equations whose solution provides the optimal nonlinearities. Numerical results based on simulation of the performance of the proposed two-sensor schemes are provided to support the analysis.