The Einstein Telescope (ET) is going to bring a revolution for the future of multimessenger 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 a 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 (SGRBs). 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 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 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 explore the 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.