We present a novel fully Bayesian analysis to constrain short gamma-ray burst (sGRB) jet structures associated with cocoon, wide-angle, and simple top-hat jet models, as well as the binary neutron star (BNS) merger rate. These constraints are made given the distance and inclination information from GW170817, observed flux of GRB 170817A, observed rate of sGRBs detected by Swift, and the neutron star merger rate inferred from LIGO's first and second observing runs. A separate analysis is conducted where a fitted sGRB luminosity function is included to provide further constraints. The jet structure models are further constrained using the observation of GW190425, and we find that the assumption that it produced a GRB 170817–like sGRB which went undetected due to the jet geometry is consistent with previous observations. We find and quantify evidence for low-luminosity and wide-angle jet structuring in the sGRB population, independently from afterglow observations, with log Bayes factors of 0.45–0.55 for such models when compared to a classical top-hat jet. Slight evidence is found for a Gaussian jet structure model over all others when the fitted luminosity function is provided, producing log Bayes factors of 0.25–0.9 ± 0.05 when compared to the other models. However, without considering GW190425 or the fitted luminosity function, the evidence favors a cocoon-like model with log Bayes factors of 0.14 ± 0.05 over the Gaussian jet structure. We provide new constraints to the BNS merger rates of 1–1300 Gpc−3 yr−1 or 2–680 Gpc−3 yr−1 when a fitted luminosity function is assumed.