Locally optimum (LO) distributed detection is considered for observations that are dependent from sensor to sensor. The necessary conditions are presented for LO distributed sensor detector designs. and a locally optimum fusion rule for an N-sensor parallel distributed detection system with dependent sensor observations is given. Specific solutions are obtained for a random signal additive noise detection problem with two sensors. These solutions indicate that the LO sensor detector nonlinearities, in general, contain a term proportional to f'/f, where f is the noise probability density function (pdf). For some non-Gaussian pdf's, the new term is significant and causes the LO sensor detector nonlinearities to be nonsymmetric even for symmetric pdfs. LO solutions are presented for finite sample sizes, and the solutions for the asymptotic case are discussed. These results are extended to yield the form of the solutions for the N-sensor LO random signal distributed detection problem that generalize the two-sensor results. >