We develop an improved empirical Bayes estimation methodology for the analysis of two-state Markov chains observed from heterogeneous individuals. First, the two transition probabilities corresponding to each chain are assumed to be drawn from a common, bivariate distribution that has beta marginals. Second, randomly missing observations are incorporated into the likelihood for the hyperparameters by efficiently summing over all possible values for the missing observations. A likelihood ratio test is used to test for dependence between the transition probabilities. Posterior distributions for the transition probabilities are also derived, as is an approximation for the equilibrium probabilities. The proposed procedures are illustrated in a numerical example and in an analysis of longitudinal store display data.