Unlike quasistatic fracture problems, the regularization of local damage models for dynamic fracture is not well understood. This knowledge gap is addressed here for two widely used regularization techniques viz. the crack band model (CBM) and rate dependent damage (RDD). Regularization effectiveness of CBM and RDD is assessed with a scalar damage model for an isotropic brittle solid across various loading rates. Results range from localized to fully diffused/branched cracks and the predicted crack velocity, fracture geometry, energy dissipation, and load displacement curves are assessed. For CBM, mesh objectivity is achieved by scaling the softening modulus with mesh size to retain constant fracture energy dissipation per unit crack length. This approach is found to be mesh independent only at lower strain rates for localized or slightly branched cracks. This is due to the linear scaling of the energy dissipation with mesh size in CBM. For branched/diffuse cracks, this scaling becomes parabolic thus making CBM inapplicable. Interestingly, for these cases not scaling the softening modulus can yield mesh independent results. For RDD, the damage initiation and evolution is made dependent on the effective strain rate, thus scaling the strength and fracture energy with strain rate, but not the softening modulus. This approach yields mesh independent results only at higher strain rates. It is shown that RDD introduces a lengthscale to the model related to the rate sensitivity parameter and wave speed. A thickening damage band is observed, wider than the mesh size. This band width is mesh independent if greater than the intrinsic lengthscale, which is when other results are mesh independent too. Thus, it is demonstrated that neither of the regularization techniques work across all loading rates and with local damage models, mesh objective prediction of dynamic fracture can be ensured only if the mesh size is fixed.