The next generation gravitational wave (GW) detectors - Einstein Telescope (ET) and Cosmic Explorer (CE), will have distance horizons up to $\mathcal{O}(10)$ Gpc for detecting binary neutron star (BNS) mergers. This makes them ideal to be used as triggers for neutrino searches from BNS mergers at the next generation neutrino detectors like IceCube-Gen2 and KM3NeT. We calculate the distance limits up to which meaningful triggers from the GW detectors can be used to minimize backgrounds and collect a good sample of neutrino events at the neutrino detectors, using the sky localization capabilities of the GW detectors. We then discuss the prospects of the next generation detectors to work in synergy to facilitate coincident neutrino detections or to constrain the parameter space in the case of non-detection of neutrinos. We find that for a typical scenario while CE and ET can provide with such detections or constraints over a timescale of $10 - 20$ years, the combination of ET+CE, owing to its good localization capabilities can lead to coincident detections or constraints at the $3 \sigma$ level within $20$ years even for the most conservative parameter choices.