  The first multi-messenger gravitational wave event has had a transformative
effect on the space of modified gravity models. In this paper we study the
enhanced tests of gravity that are possible with a future set of gravitational
wave standard siren events. We perform MCMC constraint forecasts for parameters
in Horndeski scalar-tensor theories. In particular, we focus on the
complementarity of gravitational waves with electromagnetic large-scale
structure data from galaxy surveys. We find that the addition of fifty low
redshift ($z \lesssim 0.2$) standard sirens from the advanced LIGO network
offers only a modest improvement (a factor 1.1 -- 1.3, where 1.0 is no
improvement) over existing constraints from electromagnetic observations of
large-scale structures. In contrast, high redshift (up to $z \sim 10$) standard
sirens from the future LISA satellite will improve constraints on the time
evolution of the Planck mass in Horndeski theories by a factor $\sim 5$. By
simulating different scenarios, we find this improvement to be robust to
marginalisation over unknown merger inclination angles and to variation between
three plausible models for the merger source population.
