Regular black holes (RBHs) can be seen as possible alternatives to standard black holes (BHs), since these geometries do not have a curvature singularity. As a way of improving our knowledge of such geometries, we can investigate how the astrophysical environment interacts with RBHs and compare the results with those obtained in the framework of standard BHs. We aim to study the absorption and scattering cross sections of massive scalar waves impinging on Ayón-Beato-García and Bardeen charged RBH geometries, focusing on understanding the role played by the field’s mass. Concerning the absorption spectrum, our numerical results show that the total absorption cross section decreases as we increase the field’s mass for fixed values of the BH charge. In turn, in the scattering spectrum, an increase in the mass of the field leads to wider interference widths for field velocities larger than a critical value, $v_c$, and the scattering intensity flux increases as we consider higher field mass values. We also show that scattering intensity flux profile differs for each BH spacetime as we increase the BH charge-to-mass ratio, suggesting a potentially effective mechanism for distinguishing between these geometries. Moreover, we compare our numerical results with the classical and semiclassical approximations, showing that they agree very well within the appropriate limits. We also draw comparisons with the results of the Reissner-Nordström metric. In particular, we show that the mass of the field contributes to finding situations in which the absorption and scattering spectra of regular and standard BHs are similar for arbitrary values of the field frequency and scattering angle, considering low-to near-extreme BH charges.