  In this work we show that rough stochastic differential equations (RSDEs), as
introduced by Friz, Hocquet, and L\^e (2021), are Malliavin differentiable. We
use this to prove existence of a density when the diffusion coefficients
satisfies standard ellipticity assumptions. Moreover, when the coefficients are
smooth and the diffusion coefficients satisfies a H\"ormander condition, the
density is shown to be smooth. The key ingredient is to develop a comprehensive
theory of linear rough stochastic differential equations, which could be of
independent interest.
