In this work, we study the continuum theories of dipolar-Poisson models. Both the standard dipolar-Poisson model and the dipolar-Poisson-Langevin model, which keeps the dipolar density fixed, are non-convex functionals of the scalar electrostatic potential ϕ. Applying the Legendre transform approach introduced by Maggs [Europhys. Lett. 98, 16012 (2012)], the dual functionals of these models are derived and are given by convex vector-field functionals of the dielectric displacement D and the polarization field P. We compare the convex functionals in P-space to the non-convex functionals in electric field E-space and apply them to the classic problem of the solvation of point-like ions. Since the dipolar-Poisson model does not properly describe polarization saturation, we argue that only the dipolar-Poisson-Langevin functional can be used to provide a nonlinear generalization of the harmonic polarization functional used in the theory of Marcus for the electron transfer rate to nonlinear regimes. We show that the model can be quantitatively parameterized by molecular dynamics simulations.