We perform general relativistic magnetohydrodynamic and relativistic magnetohydrodynamic simulations of weakly and highly magnetized gamma-ray burst (GRB) jets propagating in binary neutron star (BNS) merger ejecta. Using the simulations, we first find that mixing between the jet and cocoon, which is present in all types of jets, inhibits the formation of subphotospheric collisionless shocks. However, we show that a mild magnetization may lead to the formation of collisionless subshocks, which allow efficient proton acceleration. We consider shear acceleration and diffusive shock acceleration at collimation shocks, internal shocks, shock breakout, and external shocks to provide the first estimate for neutrino and cosmic-ray (CR) signals from self-consistent simulations of GRBs in BNS mergers. We find that short GRBs do not produce detectable neutrino signals with current-day facilities. Shock breakout yields ∼10 PeV neutrinos at viewing angles ∼20 , independent of the jet magnetization. However, a neutrino signal from shock breakout is well below the detection limits of current detectors. Such a signal would allow a coincident neutrino-γ-ray detection, providing a testable prediction for shock breakout as a neutrino production site. Using the numerical modeling that fits GW 170817 afterglow emission, we find that blast waves in BNS mergers can account for 5%-10% of the Galactic CR luminosity in the PeV-EeV energy range. Based on these estimates, the observed level of CR anisotropy places a constraint on the distance of the latest Galactic BNS merger to ≲3 kpc.