Meta-analyses of diagnostic test accuracy (DTA) studies have been gaining prominence in research in clinical epidemiology and health technology development. In these DTA meta-analyses, some studies may have markedly different characteristics from the others and potentially be inappropriate to include. The inclusion of these "outlying" studies might lead to biases, yielding misleading results. In addition, there might be influential studies that have notable impacts on the results. In this article, we propose Bayesian methods for detecting outlying studies and their influence diagnostics in DTA meta-analyses. Synthetic influence measures based on the bivariate hierarchical Bayesian random effects models are developed because the overall influences of individual studies should be simultaneously assessed by the two outcome variables and their correlation information. We propose four synthetic measures for influence analyses: (a) relative distance, (b) standardized residual, (c) Bayesian p-value, and (d) influence statistic on the area under the summary receiver operating characteristic curve. We also show that conventional univariate Bayesian influential measures can be applied to the bivariate random effects models, which can be used as marginal influential measures. Most of these methods can be similarly applied to the frequentist framework. We illustrate the effectiveness of the proposed methods by applying them to a DTA meta-analysis of ultrasound in screening for vesicoureteral reflux among children with urinary tract infections.