This paper proposes a strategy for regularized estimation in multi-way contingency tables, which are common in meta-analyses and multi-center clinical trials. Our approach is based on data augmentation, and appeals heavily to a novel class of Polya-Gamma distributions. Our main contributions are to build up the relevant distributional theory and to demonstrate three useful features of this data-augmentation scheme. First, it leads to simple EM and Gibbs-sampling algorithms for posterior inference, circumventing the need for analytic approximations, numerical integration, Metropolis--Hastings, or variational methods. Second, it allows modelers much more flexibility when choosing priors, which have traditionally come from the Dirichlet or logistic-normal family. For example, our approach allows users to incorporate Bayesian analogues of classical penalized-likelihood techniques (e.g. the lasso or bridge) in computing regularized estimates for log-odds ratios. Finally, our data-augmentation scheme naturally suggests a default strategy for prior selection based on the logistic-Z model, which is strongly related to Jeffreys' prior for a binomial proportion. To illustrate the method we focus primarily on the particular case of a meta-analysis/multi-center study (or a JxKxN table). But the general approach encompasses many other common situations, of which we will provide examples.