We have constructed maximally localized Wannier functions for prototype structures of solid molecular hydrogen under pressure, starting from local-density approximation and tight-binding Bloch wave functions. Each occupied Wannier function can be associated with two paired protons, defining a "Wannier molecule." The sum of the dipole moments of these "molecules" always gives the correct macroscopic polarization, even under strong compression, when the overlap between nearby Wannier functions becomes significant. We find that at megabar pressures the contributions to the dipoles arising from the overlapping tails of the Wannier functions are very large. The strong vibron infrared absorption experimentally observed in phase III, above similar to 150 GPa, is analyzed in terms of the vibron-induced fluctuations of the Wannier dipoles. We decompose these fluctuations into "static" and "dynamical" contributions, and find that at such high densities the latter term, which increases much more steeply with pressure, is dominant.