  Gravitational waves (GWs) emitted by binary sources are interesting signals
for testing gravity on cosmological scales since they allow measurements of the
luminosity distance. When followed by electromagnetic counterparts, in
particular, they enable a reconstruction of the GW-distance-redshift relation.
In the context of several modified gravity (MG) theories, even when requiring
that the speed of propagation is equal to that of light, this GW distance
differs from the standard electromagnetic luminosity distance due to the
presence of a modified friction in the GW propagation. The very same source of
this friction, which is the running of an effective Planck mass, also affects
the scalar sector generating gravitational slip, i.e. a difference between the
scalar potentials, an observable that can be inferred from large-scale
structure (LSS) probes. In this work, we use a model within effective field
theories for dark energy to exemplify precisely the fact that, at the linear
perturbation level, parametrizing a single function is already enough to
generate simultaneous deviations in the GW distance and the slip. By simulating
multi-messenger GW events that might be detected by the Einstein Telescope in
the future, we compare the constraining power of the two observables on this
single degree of freedom. We then combine forecasts of a Euclid-like survey
with GW simulations, coming to the conclusion that, when using Planck data to
better constrain the cosmological parameters, those future data on the scalar
and tensor sectors are competitive to probe such deviations from General
Relativity, with LSS giving stronger (but more model-dependent) results than
GWs.
