We address the problem of distributed estimation from dependent observations involving two sensors that collect observations of the same nonrandom location parameter THETA in additive noise. We consider two cases of interest, the case of independent observations across sensors and the case of correlated observations across sensors. The estimation schemes of the sensors are chosen so as to minimize a common cost function consisting of the weighted sum of the mean square errors of the estimates from the two sensors and the mean square of their difference. The observations of the two sensors are modeled as two MU - dependent or PHI mixing sequences. The correlation between the two observation sequences is also characterized by an p-dependent or PHI mixing sequence. Because high-order statistics of dependent observations are generally difficult to characterize, maximum-likehood estimates may be impossible to derive or implement; instead, suboptimal estimates which use memoryless nonlinearities g_k (DOT) (i.e. nonlinear functions of observations,) for k = 1,2, are employed by the two sensors. With this structure in each sensor, minimizing the above cost function with respect to the estimates is equivalent to minimizing it with respect to the nonlinearities g_k (DOT), which results in linear integral equations. If we solve these integral equations, we obtain optimal nonlinearities within this suboptimal scheme. Examples for m - dependent Cauchy noise are provided in support of our analysis.