The joint estimation of multiple graphical models from high-dimensional data has been studied in the statistics and machine learning literature, owing to its importance in diverse fields including molecular biology, neuroscience, and the social sciences. We pro- pose a Bayesian approach that decomposes the model parameters across multiple graphi- cal models into shared components across subsets of models and edges, and idiosyncratic components. This approach leverages a novel multivariate prior distribution, coupled with a jointly convex regression-based pseudo-likelihood that enables fast computation using a robust and efficient Gibbs sampling scheme. We establish strong posterior consistency for model selection under high-dimensional scaling, with the number of variables growing exponentially as a function of the sample size. Lastly, we demonstrate the efficiency of the proposed approach in borrowing strength across models to identify shared edges using both synthetic and real data.