  The key probes of the growth of large-scale structure are its rate $f$ and
amplitude $\sigma_8$. Redshift space distortions in the galaxy power spectrum
allow us to measure only the combination $f\sigma_8$, which can be used to
constrain the standard cosmological model or alternatives. By using
measurements of the galaxy-galaxy lensing cross-correlation spectrum or of the
galaxy bispectrum, it is possible to break the $f\sigma_8$ degeneracy and
obtain separate estimates of $f$ and $\sigma_8$ from the same galaxy sample.
Currently there are only a handful of such separate measurements, but even this
allows for improved constraints on cosmological models. We explore how having a
larger and more precise sample of such measurements in the future could
constrain further cosmological models. We consider what can be achieved by a
future nominal sample that delivers a $\sim 1\%$ constraint on $f$ and
$\sigma_8$ separately, compared to the case with a similar precision on the
combination $f\sigma_8$. For the six cosmological parameters of $\Lambda$CDM,
we find improvements of $\sim\! 5$--$50\%$ on their constraints. For modified
gravity models in the Horndeski class, the improvements on these standard
parameters are $\sim\! 0$--$15\%$. However, the precision on the sum of
neutrino masses improves by 65\% and there is a significant increase in the
precision on the background and perturbation Horndeski parameters.
