  With the measurement of the electromagnetic (EM) counterpart, a gravitational
wave (GW) event could be treated as a standard siren. As a novel cosmological
probe, GW standard sirens will bring significant implications for cosmology. In
this paper, by considering the coincident detections of GW and associated
$\gamma$ ray burst (GRB), we find that only about 400 GW bright standard sirens
from binary neutron star mergers could be detected in a 10-year observation of
the Einstein Telescope and the THESEUS satellite mission. Based on this mock
sample, we investigate the implications of GW standard sirens on the
interaction between dark energy and dark matter. In our analysis, four viable
interacting dark energy (IDE) models, with interaction forms $Q=3\beta H
\rho_{\mathrm{de}}$ and $Q=3 \beta H \rho_{\mathrm{c}}$, are considered.
Compared with the traditional EM observational data such as CMB, BAO, and SN
Ia, the combination of both GW and EM observations could effectively break the
degeneracies between different cosmological parameters and provide more
stringent cosmological fits. We find that the GW data could play a more
important role for determining the interaction in the models with $Q=3 \beta H
\rho_{\mathrm{c}}$, compared with the models with $Q=3\beta H
\rho_{\mathrm{de}}$. We also show that constraining IDE models with mock GW
data based on different fiducial $H_0$ values yield different results,
indicating that accurate determination of $H_0$ is significant for exploring
the interaction between dark energy and dark matter.
