  Scattering transforms have been successfully used to describe dust
polarization for flat-sky images. This paper expands this framework to noisy
observations on the sphere with the aim of obtaining denoised Stokes Q and U
all-sky maps at 353GHz, as well as a non-Gaussian model of dust polarization,
from the Planck data. To achieve this goal, we extend the computation of
scattering coefficients to the Healpix pixelation and introduce
cross-statistics that allow us to make use of half-mission maps as well as the
correlation between dust temperature and polarization. Introducing a general
framework, we develop an algorithm that uses the scattering statistics to
separate dust polarization from data noise. The separation is validated on mock
data, before being applied to the SRoll2 Planck maps at N_side = 256. The
validation shows that the statistics of the dust emission, including its
non-Gaussian properties, are recovered until l < 700, where, at high Galactic
latitudes, the dust power is smaller than that of the dust by two orders of
magnitude. On scales where the dust power is lower than one tenth of that of
the noise, structures in the output maps have comparable statistics but are not
spatially coincident with those of the input maps. Our results on \Planck\ data
are significant milestones opening new perspectives for statistical studies of
dust polarization and for the simulation of Galactic polarized foregrounds. The
Planck denoised maps is available (see
http://sroll20.ias.u-psud.fr/sroll40_353_data.html) together with results from
our validation on mock data, which may be used to quantify uncertainties.
