We examine the dust geometry and Ly alpha scattering in the galaxies of the Lyman Alpha Reference Sample (LARS), a set of 14 nearby (0.02 < z < 0.2) Ly alpha-emitting and starbursting systems with Hubble Space Telescope Ly alpha, H alpha, and H beta imaging. We find that the global dust properties determined by line ratios are consistent with other studies, with some of the LARS galaxies exhibiting clumpy dust media, while others of them show significantly lower Ly alpha emission compared to their Balmer decrement. With the LARS imaging, we present Ly alpha/H alpha and H alpha/H beta maps with spatial resolutions as low as similar to 40 pc, and use these data to show that in most galaxies, the dust geometry is best modeled by three distinct regions: a central core where dust acts as a screen, an annulus where dust is distributed in clumps, and an outer envelope where Ly alpha photons only scatter. We show that the dust that affects the escape of Ly alpha is more restricted to the galaxies' central regions, while the larger Ly alpha halos are generated by scattering at large radii. We present an empirical modeling technique to quantify how much Ly alpha scatters in the halo, and find that this characteristic scattering distance correlates with the measured size of the Ly alpha halo. We note that there exists a slight anti-correlation between the scattering distance of Ly alpha and global dust properties.