We use the gravitational wave (GW) events GW170817 and GW190521, together with their proposed electromagnetic counterparts, to constrain cosmological parameters and theories of gravity beyond General Relativity (GR). In particular we consider models with a time-varying Planck mass, large extra-dimensions, and a phenomenological parametrization covering several beyond-GR theories. In all three cases, this introduces a friction term into the GW propagation equation, effectively modifying the GW luminosity distance. We set constraints on ACDM parameters and GR deviation parameters using two sets of priors on the Hubble constant and matter energy density. With priors set to the measured Planck's mission values, we find that with the inclusion of GW190521, the two GR deviation parameters constraints improve by a factor similar to 10. We report a number of space-time dimensions compatible with 4 with an precision of 2.5% (at 95% CL) and an upper limit to the variation of Newton's constant at the epoch of GW170817 of < 20%. With wide priors on the Hubble constant and matter energy density, we show that it is still possible to constrain the ACDM parameters and GR deviation parameters conjointly from GW170817 and GW190521, obtaining constraints on GR deviation parameters which are a factor 2-6 worse than the results using restricted priors on ACDM parameters.