  The Einstein Telescope (ET) is going to bring a revolution for the future of
multi-messenger astrophysics. In order to detect the counterparts of binary
neutron star (BNS) mergers at high redshift, the high-energy observations will
play a crucial role. Here, we explore the perspectives of ET, as single
observatory and in a network of gravitational-wave (GW) detectors, operating in
synergy with future $\gamma$-ray and X-ray satellites. We predict the
high-energy emission of BNS mergers and its detectability in a theoretical
framework which is able to reproduce the properties of the current sample of
observed short GRBs (SGRB). We estimate the joint GW and high-energy detection
rate for both the prompt and afterglow emissions, testing several combinations
of instruments and observational strategies. We find that the vast majority of
SGRBs detected in $\gamma$-rays will have a detectable GW counterpart; the
joint detection efficiency approaches $100\%$ considering a network of third
generation GW observatories. The probability of identifying the electromagnetic
counterpart of BNS mergers is significantly enhanced if the sky localisation
provided by GW instruments is observed by wide field X-ray monitors. We
emphasize that the role of the future X-ray observatories will be very crucial
for the detection of the fainter emission outside the jet core, which will
allow us to probe the yet unexplored population of low-luminosity SGRBs in the
nearby Universe, as well as to unveil the nature of the jet structure and the
connections with the progenitor properties.
