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With the recent development of assays that detect multiple diseases, it is now common to test biospecimen pools for multiple infections simultaneously. Recent work has developed an expectation\u2013maximization algorithm to estimate the prevalence of two infections using a two\u2010stage, Dorfman\u2010type testing algorithm motivated by current screening practices for chlamydia and gonorrhea in the USA. In this article, we have the same goal but instead take a more flexible Bayesian approach. Doing so allows us to incorporate information about assay uncertainty during the testing process, which involves testing both pools and individuals, and also to update information as individuals are tested. Overall, our approach provides reliable inference for disease probabilities and accurately estimates assay sensitivity and specificity even when little or no information is provided in the prior distributions. We illustrate the performance of our estimation methods using simulation and by applying them to chlamydia and gonorrhea data collected in Nebraska. 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