  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.
