Gravitational-wave sources can serve as standard sirens to probe cosmology by measuring their luminosity distance and redshift. Such standard sirens are also useful to probe theories beyond general relativity with a modified gravitational-wave propagation. Many of previous studies on the latter assume multi-messenger observations so that the luminosity distance can be measured with gravitational waves while the redshift is obtained by identifying sources host galaxies from electromagnetic counterparts. Given that gravitational-wave events of binary neutron star coalescences with associated electromagnetic counterpart detections are expected to be rather rare, it is important to examine the possibility of using standard sirens with gravitational-wave observations alone to probe gravity. In this paper, we achieve this by extracting the redshift from the tidal measurement of binary neutron stars that was originally proposed within the context of gravitational-wave cosmology (another approach is to correlate dark sirens with galaxy catalogs that we do not consider here). We consider not only observations with ground-based detectors (e.g. Einstein Telescope) but also multi-band observations between ground-based and space-based (e.g. DECIGO) interferometers. We find that such multi-band observations with the tidal information can constrain a parametric non-Einsteinian deviation in the luminosity distance (due to the modified friction in the gravitational wave evolution) more stringently than the case with electromagnetic counterparts by a factor of a few. We also map the above-projected constraints on the parametric deviation to those on specific theories and phenomenological models beyond general relativity to put the former into context.
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