To evaluate the probabilities of a disease state, ideally all subjects in a study should be diagnosed by a definitive diagnostic or gold standard test. However, since definitive diagnostic tests are often invasive and expensive, it is generally unethical to apply them to subjects whose screening tests are negative. In this article, we consider latent class models for screening studies with two imperfect binary diagnostic tests and a definitive categorical disease status measured only for those with at least one positive screening test. Specifically, we discuss a conditional-independent and three homogeneous conditional-dependent latent class models and assess the impact of misspecification of the dependence structure on the estimation of disease category probabilities using frequentist and Bayesian approaches. Interestingly, the three homogeneous-dependent models can provide identical goodness-of-fit but substantively different estimates for a given study. However, the parametric form of the assumed dependence structure itself is not 'testable' from the data, and thus the dependence structure modeling considered here can only be viewed as a sensitivity analysis concerning a more complicated non-identifiable model potentially involving a heterogeneous dependence structure. Furthermore, we discuss Bayesian model averaging together with its limitations as an alternative way to partially address this particularly challenging problem. The methods are applied to two cancer screening studies, and simulations are conducted to evaluate the performance of these methods. In summary, further research is needed to reduce the impact of model misspecification on the estimation of disease prevalence in such settings.