Reliability assessment in presence of epistemic uncertainty leads to consider the failure probability as a quantity depending on the state of knowledge about uncertain input parameters. The input joint distribution is often learnt from a small-sized dataset provided by operating experience. The computed failure probability depends on the estimated marginal distributions and the estimated copula distribution. This paper develops a reliability-oriented sensitivity analysis procedure in order to measure the influence exerted by the data-driven modeling of both the margins and the copula. The proposed methodology is validated for both deterministic and stochastic reliability methods through an extensive simulation study including several analytical performance functions as well as a real-life simulation code dealing with the buckling of a composite laminate plate.