  We present a novel fully Bayesian analysis to constrain short gamma-ray burst
jet structures associated with cocoon, wide-angle and simple top-hat jet
models, as well as the binary neutron star merger rate. These constraints are
made given the distance and inclination information from GW170817, observed
flux of GRB170817A, observed rate of short gamma-ray bursts 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 short gamma-ray
burst 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 GRB170817-like short gamma-ray
burst that went undetected due to the jet geometry is consistent with previous
observations. We find and quantify evidence for low luminosity and wide-angled
jet structuring in the short gamma-ray burst 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 $0.25{-}0.9\pm0.05$ when
compared to the other models. However without considering GW190425 or the
fitted luminosity function, the evidence favours a cocoon-like model with log
Bayes factors of $0.14\pm0.05$ over the Gaussian jet structure. We provide new
constraints to the binary neutron star merger rates of
$1{-}1300$Gpc$^{-3}$yr$^{-1}$ or $2{-}680$Gpc$^{-3}$yr$^{-1}$ when a fitted
luminosity function is assumed.
