Spatial statistical models are applied in many problems for which dependence in observed random variables is not easily explained by a direct scientific mechanism. In such situations there may be a latent spatial process that acts to produce the observed spatial pattern. Scientific interest often centers on the latent process and the degree of spatial dependence that characterizes it. Such latent processes may be thought of as spatial mixing distributions. We present methods for the specification of flexible joint distributions to model spatial processes through multi-parameter exponential family conditional distributions. One approach to the analysis of these models is Monte Carlo maximum likelihood, and an approach based on independence pseudo-models is presented for formulating importance sampling distributions that allow such an analysis. The methods developed are applied to a problem of forest-health monitoring, where the numbers of affected trees in spatial field plots are modeled using a spatial beta-binomial mixture.