The Bayesian arm-based (AB) network meta-analysis (NMA) model require researchers to specify a prior distribution for the covariance matrix of the treatment-specific event rates in a transformed scale. The conjugate prior for the covariance matrix, the inverse-Wishart (IW) distribution, has several limitations. It generally leads to underestimation of correlations between treatment-specific log-odds, which are critical for borrowing strength across treatment arms to estimate treatment effects efficiently and to reduce potential bias. Alternatively, several separation strategies (i.e., separate priors on variances and correlations) can be considered. To study the IW prior's impact on NMA results and compare it with separation strategies, we did simulation studies under different missing-treatment mechanisms. A separation strategy with appropriate priors for the correlation matrix and variances performs better than the IW prior, and should be recommended as the default vague prior in the AB NMA approach. Finally, we re-analyzed three case studies and illustrated the importance, when performing AB-NMA, of sensitivity analyses with different prior specifications on variances.