Gamma-ray burst (GRB) prompt emission is highly beamed, and understanding the jet geometry and beaming configuration can provide information on the poorly understood central engine and circum-burst environment. Prior to the advent of gravitational-wave astronomy, astronomers relied on observations of jet breaks in the multi-wavelength afterglow to determine the GRB opening angle, since the observer's viewing angle relative to the system cannot be determined from the electromagnetic data alone. Gravitational-wave observations, however, provide an independent measurement of the viewing angle. We describe a Bayesian method for determining the geometry of short GRBs using coincident electromagnetic and gravitational-wave observations. We demonstrate how an ensemble of multi-messenger detections can be used to measure the distributions of the jet energy, opening angle, Lorentz factor, and angular profile of short GRBs; we find that for a population of 100 such observations, we can constrain the mean of the opening angle distribution to within 10 • regardless of the angular emission profile. Conversely, the constraint on the energy distribution depends on the shape of the profile, which can be distinguished.