  The next generation of detectors will detect gravitational waves from binary
neutron stars at cosmological distances, for which around a thousand
electromagnetic follow-ups may be observed per year. So far, most work devoted
to the expected cosmological impact of these standard sirens employed them only
as distance indicators. Only recently their use as tracers of clustering,
similar to what already proposed for supernovae, has been studied. Focusing on
the expected specifications of the Einstein Telescope (ET), we forecast here
the performance on cosmological parameters of future standard sirens as both
distance and density indicators, with emphasis on the linear perturbation
growth index and on spatial curvature. We improve upon previous studies in a
number of ways: a more detailed analysis of available telescope time, the
inclusion of more cosmological and nuisance parameters, the Alcock-Paczynski
correction, the use of sirens also as both velocity and density tracers, and a
more accurate estimation of the distance posterior. We find that the analysis
of the clustering of sirens improves the constraints on $H_0$ by 30% and on
$\Omega_{k0}$ by over an order of magnitude, with respect to their use merely
as distance indicators. With 5 years of joint ET and Rubin Observatory
follow-ups we could reach precision of 0.1 km/s/Mpc in $H_0$ and 0.02 in
$\Omega_{k0}$ using only data in the range $0<z<0.5$. We also find that the use
of sirens as tracers of density, and not only velocity, yields good
improvements on the growth of structure constraints.
