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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.