Bayesian statistical approaches to mixed treatment comparisons (MTCs) are becoming more popular because of their flexibility and interpretability. Many randomized clinical trials report multiple outcomes with possible inherent correlations. Moreover, MTC data are typically sparse (although richer than standard meta-analysis, comparing only two treatments), and researchers often choose study arms based upon which treatments emerge as superior in previous trials. In this paper, we summarize existing hierarchical Bayesian methods for MTCs with a single outcome and introduce novel Bayesian approaches for multiple outcomes simultaneously, rather than in separate MTC analyses. We do this by incorporating partially observed data and its correlation structure between outcomes through contrast-based and arm-based parameterizations that consider any unobserved treatment arms as missing data to be imputed. We also extend the model to apply to all types of generalized linear model outcomes, such as count or continuous responses. We offer a simulation study under various missingness mechanisms (e.g., missing completely at random, missing at random, and missing not at random) providing evidence that our models outperform existing models in terms of bias, mean squared error, and coverage probability then illustrate our methods with a real MTC dataset. We close with a discussion of our results, several contentious issues in MTC analysis, and a few avenues for future methodological development.