Resonant scatterers such as hydrogen adatoms can strongly enhance the low-energy density of states in graphene. Here, we study the impact of these impurities on electronic screening. We find a two-faced behavior: Kubo formula calculations reveal an increased dielectric function ε upon creation of midgap states but no metallic divergence of the static ε at small momentum transfer q→0. This bad metal behavior manifests also in the dynamic polarization function and can be directly measured by means of electron energy loss spectroscopy. A new length scale lc beyond which screening is suppressed emerges, which we identify with the Anderson localization length.