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The Einstein Telescope (ET) is going to bring a revolution for the future ofmulti-messenger astrophysics. In order to detect the counterparts of binaryneutron star (BNS) mergers at high redshift, the high-energy observations willplay a crucial role. Here, we explore the perspectives of ET, as singleobservatory and in a network of gravitational-wave (GW) detectors, operating insynergy with future $\gamma$-ray and X-ray satellites. We predict thehigh-energy emission of BNS mergers and its detectability in a theoreticalframework which is able to reproduce the properties of the current sample ofobserved short GRBs (SGRB). We estimate the joint GW and high-energy detectionrate for both the prompt and afterglow emissions, testing several combinationsof instruments and observational strategies. We find that the vast majority ofSGRBs detected in $\gamma$-rays will have a detectable GW counterpart; thejoint detection efficiency approaches $100\%$ considering a network of thirdgeneration GW observatories. The probability of identifying the electromagneticcounterpart of BNS mergers is significantly enhanced if the sky localisationprovided by GW instruments is observed by wide field X-ray monitors. Weemphasize that the role of the future X-ray observatories will be very crucialfor the detection of the fainter emission outside the jet core, which willallow us to probe the yet unexplored population of low-luminosity SGRBs in thenearby Universe, as well as to unveil the nature of the jet structure and theconnections with the progenitor properties.

Perspectives for multi-messenger astronomy with the next generation of gravitational-wave detectors and high-energy satellites