Existing methods for meta-analysis of diagnostic test accuracy focus primarily on a single index test. We propose models for the joint meta-analysis of studies comparing multiple index tests on the same participants in paired designs. These models respect the grouping of data by studies, account for the within-study correlation between the tests' true-positive rates (TPRs) and between their false-positive rates (FPRs) (induced because tests are applied to the same participants), and allow for between-study correlations between TPRs and FPRs (such as those induced by threshold effects). We estimate models in the Bayesian setting. We demonstrate using a meta-analysis of screening for Down syndrome with two tests: shortened humerus (arm bone), and shortened femur (thigh bone). Separate and joint meta-analyses yielded similar TPR and FPR estimates. For example, the summary TPR for a shortened humerus was 35.3% (95% credible interval (CrI): 26.9, 41.8%) versus 37.9% (27.7, 50.3%) with joint versus separate meta-analysis. Joint meta-analysis is more efficient when calculating comparative accuracy: the difference in the summary TPRs was 0.0% (-8.9, 9.5%; TPR higher for shortened humerus) with joint versus 2.6% (-14.7, 19.8%) with separate meta-analyses. Simulation and empirical analyses are needed to refine the role of the proposed methodology.