This paper considers Bayesian models for inference on unknown spatial weights in a spatial error model. We study two different assumptions that ensure identification, (i) symmetric spatial weights, and (ii) triangular systems. We propose Bayesian methods for inference under these assumptions and study the finite sample performance of these estimators. The results indicate that the Bayesian estimator performs better than existing methods in small samples. We illustrate the symmetric case with a study of spatial dependence in healthcare spending among federal states in Germany. The results show that there is strong and systematic spatial dependence. However, the estimated spatial structure differs greatly from structures defined by geography. This implies that spatial spillovers are rather driven by technology diffusion and socio-cultural linkages between regions rather than geographic distances.