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 353 GHz, 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 max ∼ 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 will be made available together with results from our validation on mock data, which may be used to quantify uncertainties.