Nonlocal continua have been proposed to offer a more realistic model for the electrostatic response of solutions such as the electrolyte solvents prominent in biology and electrochemistry. In this work, we review three nonlocal models based on the Landau-Ginzburg framework which have been proposed but not directly compared previously, due to different expressions of the nonlocal constitutive relationship. To understand the relationships between these models and the underlying physical insights from which they are derive, we situate these models into a single, unified Landau-Ginzburg framework. One of the models offers the capacity to interpret how temperature changes affect dielectric response, and we note that the variations with temperature are qualitatively reasonable even though predictions at ambient temperatures are not quantitatively in agreement with experiment. Two of these models correctly reproduce overscreening (oscillations between positive and negative polarization charge densities), and we observe small differences between them when we simulate the potential between parallel plates held at constant potential. These computations require reformulating the two models as coupled systems of local partial differential equations (PDEs), and we use spectral methods to discretize both problems. We propose further assessments to discriminate between the models, particularly in regards to establishing boundary conditions and comparing to explicit-solvent molecular dynamics simulations.