We have considered the electrostatic field distribution near a soft permeable interface in an aqueous solution, taking into account the spatial variation of the dielectric permittivity within the framework of a nonlocal electrostatic theory. We have analyzed the effect of the interface thickness as well as different combinations of the surface electric charge and dipole densities. We have shown that the electric field potential is described in such systems by two exponential terms, one of which is determined by Debye screening, and the other by the nonlocal polarization of the aqueous solution. For certain combinations of the electric charge and dipole densities the potential can become nonmonotonic. We have shown that, in addition to the electric charge and dipole densities, a significant parameter that determines the electric field distribution in electric double layers is the interface thickness. We discuss how the results obtained can assist in understanding model lipid/protein systems, measurements of surface potentials, and surfaces of biological membranes.