Reversible molecular-dynamics (MD) simulations have been carried out on simple models for dispersions of circular Laponite clay platelets to investigate the local structure on a mesoscopic scale. The platelets carry discrete charged sites interacting via a screened Coulomb potential. In model A all surface sites have identical negative charge, while model B also includes rim charges of opposite sign. These two models were used in a series of simulations in the semidilute regime, and for three values of the Debye screening length. The structure of the dispersions is characterized by translational and orientational pair distribution functions, and by the corresponding structure factors. Qualitative differences in the pair structure arising from variations in concentration and screening length lead to a tentative identification of sol, gel, and crystal phases. The rim charges have a dramatic effect on the local structure in the strong screening regime, leading to T-shaped pair configuration and clustering of the platelets at low clay concentrations, and at higher concentrations to a space-filling "house-of-cards" structure.