Ultra-high energy cosmic rays (UHE CRs) interacting with the cosmic radiation background produce two cosmogenic messengers: neutrinos with energies in the EeV range and gamma rays accumulating in the GeV-TeV range. The most optimistic scenario for cosmogenic fluxes assumes the dominance of protons above the Greisen-Zatsepin-Kuzmin threshold of resonant scattering with photons in the cosmic microwave background. Whereas these optimistic cosmogenic fluxes are testable with present observatories, the corresponding predictions of heavier UHE CR composition models are orders of magnitude smaller, falling within the domain of more sensitive future detectors. In this study we use the latest results of the Pierre Auger observatory for the UHE CR spectrum and chemical composition to derive conservative lower limits on the cosmogenic neutrino and gamma ray fluxes. We investigate the prospects and requirements of future large-scale neutrino and CR observatories to observe these fluxes.