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 γ -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 γ -rays will have a detectable GW counterpart; the joint detection e ffi ciency 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 localization 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.