Bivariate Poisson models are suitable for modelling paired count data. Nonetheless, the bivariate Poisson model does not allow for negative dependence structure, therefore it is necessary to consider alternatives, which can produce both positive and negative dependence. A natural approach is to use copulas to create different bivariate discrete distributions. Although such models exist in the literature, the problem of selection an appropriate copula has been overlooked so far. Various copulas lead to different structure, any copula misspecification can render the inference useless. In this work, we consider bivariate Poisson models generated with a copula and investigate its robustness under outliers contamination and model misspecification. Special emphasis is given on the robustness of copula related parameters.