The joint spatial distribution of two count outcomes (eg, counts of two diseases) is usually studied using a Poisson shared component model (P-SCM), which uses geographically structured latent variables to model spatial variations that are specific and shared by both outcomes. In this model, the correlation between the outcomes is assumed to be fully accounted for by the latent variables. However, in this article, we show that when the outcomes have an unknown number of cases in common, the bivariate counts exhibit a positive "residual" correlation, which the P-SCM wrongly attributes to the covariance of the latent variables, leading to biased inference and degraded predictive performance. Accordingly, we propose a new SCM based on the Bivariate-Poisson distribution (BP-SCM hereafter) to study such correlated bivariate data. The BP-SCM decomposes each count into counts of common and distinct cases, and then models each of these three counts (two distinct and one common) using Gaussian Markov Random Fields. The model is formulated in a Bayesian framework using Hamiltonian Monte Carlo inference. Simulations and a real-world application showed the good inferential and predictive performances of the BP-SCM and confirm the bias in P-SCM. BP-SCM provides rich epidemiological information, such as the mean levels of the unknown counts of common and distinct cases, and their shared and specific spatial variations.