Studies of dark energy at advanced gravitational-wave (GW) interferometers normally focus on the dark energy equation of state w_{\rm DE}(z). However, modified gravity theories that predict a non-trivial dark energy equation of state generically also predict deviations from general relativity in the propagation of GWs across cosmological distances, even in theories where the speed of gravity is equal to c. We find that, in generic modified gravity models, the effect of modified GW propagation dominates over that of w_{\rm DE}(z), making modified GW propagation a crucial observable for dark energy studies with standard sirens. We present a convenient parametrization of the effect in terms of two parameters (\Xi_0,n), analogue to the (w_0,w_a) parametrization of the dark energy equation of state, and we give a limit from the LIGO/Virgo measurement of H0 with the neutron star binary GW170817. We then estimate the sensitivity of the Einstein Telescope (ET) to \Xi_0, combining standard sirens with other cosmological datasets and performing a Markov Chain Monte Carlo analysis. We discuss the prediction of a specific nonlocal modification of gravity, recently developed by our group, and we show that its predictions are within the reach of ET. Modified GW propagation also affects the GW transfer function, and therefore modifies the present-day energy density of stochastic backgrounds of GWs, as well as the tensor contribution to the ISW effect.
