Because of the high cost and time constraints for clinical trials, researchers often need to determine the smallest sample size that provides accurate inferences for a parameter of interest. Although most experimenters have employed frequentist sample-size determination methods, the Bayesian paradigm offers a wide variety of sample-size determination methodologies. Bayesian sample-size determination methods are becoming increasingly more popular in clinical trials because of their flexibility and easy interpretation inferences. Recently, Bayesian approaches have been used to determine the sample size of a single Poisson rate parameter in a clinical trial setting. In this paper, we extend these results to the comparison of two Poisson rates and develop methods for sample-size determination for hypothesis testing in a Bayesian context. We have created functions in R to determine the parameters for the conjugate gamma prior and calculate the sample size for the average length criterion and average power methods. We also provide two examples that implement our sample-size determination methods using clinical data.