Bayesian inference procedures for the parameters of the multivariate random effects model are derived under the assumption of an elliptically contoured distribution when the Berger and Bernardo reference and the Jeffreys priors are assigned to the model parameters. A new numerical algorithm for drawing samples from the posterior distribution is developed, which is based on the hybrid Gibbs sampler. The new approach is compared to the two Metropolis -Hastings algorithms previously derived in the literature via an extensive simulation study. The findings are applied to a Bayesian multivariate meta -analysis, conducted using the results of ten studies on the effectiveness of a treatment for hypertension. The analysis investigates the treatment effects on systolic and diastolic blood pressure. The second empirical illustration deals with measurement data from the CCAUV.V-K1 key comparison, aiming to compare measurement results of sinusoidal linear accelerometers at four frequencies.