Multilevel (or variance components) models have been applied in many areas of regional science, epidemiology and polimetrics. They are most often used to model treatment nonstationarity in policy regimes, a form of spatial process heterogeneity. Multilevel models with spatially-correlated components are increasingly used to model the presence of both spatial heterogeneity and spatial dependence. Previous treatments typically focus on studying a single model specification in detail, deriving its estimators, and demonstrating its properties in a case study. Instead, a general approach is possible. In this paper, a generic Gibbs sampler for spatially-correlated multilevel models is developed and its empirical properties examined in model of cancer screening. Critically, there is tension between spatial dependence and spatial heterogeneity terms in the model. This results in apparent ``growth'' in some multilevel models where the dependence terms overpower the effects of estimate regularization. This results in locally-smoothed estimates, which, depending on the strength of spatial dependence, may be both larger and more precise than their corresponding single-level standard linear models. Thus, shrinkage always applies, but not always from the standard single-level linear model. Due to the complexity of these tradeoffs between model fit, dependence, and heterogeneity terms, this formal attention is critical to building systematic and transferable understanding.