We develop a theory for investigating atomic-scale dielectric permittivity profiles across interfaces between insulators. A local susceptibility chi(x;omega) is introduced to describe variations of the dielectric response over length scales of the order of interatomic distances. The nonlocality of the microscopic susceptibility tensor chi(ij)(r,r';omega) occurs at smaller distances and therefore does not intervene in our formulation. The local permittivity is obtained from the microscopic charge density induced by an applied electric field. We show that the permittivity can conveniently be analyzed in terms of maximally localized Wannier functions. In this way, we can relate variations of the microscopic dielectric response to specific features of the local bonding arrangement. In addition to a continuous description in terms of the local permittivity, we introduce an alternative scheme based on discrete polarizabilities. In the latter case, electronic polarizabilities alpha((n))(elec) are obtained in terms of the displacements of maximally localized Wannier functions, while ionic polarizabilities alpha((I))(ion) are determined from the induced ionic displacements and the corresponding dynamical charges. The potential of our scheme is illustrated through applications to two systems of technological interest. First, we consider the permittivity of Si slabs of finite thickness. Our approach indicates that the local permittivity in the slab interior approaches the corresponding value for bulk Si within a few atomic layers from the surface. Therefore, the decrease of the average slab permittivity with thickness originates from the lower permittivity of the outer planes and the increasing surface-to-volume ratio. Second, we address the dielectric permittivity across the Si(100)-SiO2 interface. Using two distinct interface models, we are able to show that the dielectric transition from the silicon to the oxide occurs within a width of only a few angstrom. The polarizability associated to intermediate oxidation states of Si is found to be enhanced with respect to bulk SiO2, resulting in a larger permittivity of the interfacial suboxide layer with respect to the stoichiometric oxide.