We address two problems of memoryless distributed dependent observations across time and sensors. In the first problem, the observation sequence of each sensor consists of common weak signal in additive dependent noise with stationary univariate and second-order joint densities; here the objective of the sensors is to cooperatively detect the presence of a weak signal. In the second problem, the observation sequence of each sensor is characterized by its stationary univariate and second-order pint densities; here the objective of the sensors is to cooperatively discriminate between two arbitrary such sequences of observations. For both problems, the analysis and design are based on a common large sample size. The dependence acms time and sensors is modeled by m-dependent, f-mixing, or p-mixing processes. The performance of the two-sensor configuration for each problem is measured by an average cost, which couples the decisions of the sensors. The design criteria for the test statistics of the sensors, which consist of sums of memoryless nonlinearities, are established by using two-dimensional Chemoff bounds on the associated error probabilities involved in the average cost. The optimal nonlinearities are obtain as the solutions of linear coupled or uncoupled integral equations. Numerical results for specific cases of practical interest show that the performance of the proposed scheme is superior to the one that ignores the dependence across time and/or sensors for each of the two problems.