  Measurements of the growth rate of structure, $f\sigma_8$ in the low-redshift
Universe allow stringent tests of the cosmological model. In this work, we
provide new constraints on $f\sigma_{8}$ at an effective redshift of $z=0.03$
using the combined density and velocity fields measured by the 2MTF and 6dFGSv
surveys. We do this by applying a new estimator of the redshift-space density
and momentum (density-weighted velocity) power spectra, developed in the first
paper of this series, to measured redshifts and peculiar velocities from these
datasets. We combine this with models of the density and momentum power spectra
in the presence of complex survey geometries and with an ensemble of simulated
galaxy catalogues that match the survey selection functions and galaxy bias. We
use these simulations to estimate the errors on our measurements and identify
possible systematics. In particular, we are able to identify and remove biases
caused by the non-Gaussianity of the power spectra by applying the Box-Cox
transformation to the power spectra prior to fitting. After thorough validation
of our methods we recover a constraint of
$f\sigma_8(z_{\mathrm{eff}}=0.03)=0.404^{+0.082}_{-0.081}$ from the combined
2MTF and 6dFGSv data. This measurement is fully consistent with the
expectations of General Relativity and the $\Lambda$ Cold Dark Matter
cosmological model. It is also comparable and complementary to constraints
using different techniques on similar data, affirming the usefulness of our
method for extracting cosmology from velocity fields.
