Short gamma-ray bursts are believed to be produced by both binary neutron star (BNS) and neutron star-black hole (NSBH) mergers. We use current estimates for the BNS and NSBH merger rates to calculate the fraction of observable short gamma-ray bursts produced through each channel. This allows us to constrain merger rates of a BNS to R-BNS = 384(-213)(+431) Gpc(-3) yr(-1) (90% credible interval), a 16% decrease in the rate uncertainties from the second Laser Interferometer Gravitational Wave Observatory (LIGO)-Virgo Gravitational-Wave Transient Catalog. Assuming a top-hat emission profile with a large Lorentz factor, we constrain the average opening angle of gamma-ray burst jets produced in BNS mergers to approximate to 15 degrees. We also measure the fraction of BNS and NSBH mergers that produce an observable short gamma-ray burst to be 0.02(-0.01)(+0.02) and 0.01 +/- 0.01, respectively, and find that greater than or similar to 40% of BNS mergers launch jets (90% confidence). We forecast constraints for future gravitational-wave detections given different modeling assumptions, including the possibility that BNS and NSBH jets are different. With 24 BNS and 55 NSBH observations, expected within six months of the LIGO-Virgo-Kamioka Gravitational Wave Detector network operating at design sensitivity, it will be possible to constrain the fraction of BNS and NSBH mergers that launch jets with 10% precision. Within a year of observations, we can determine whether the jets launched in NSBH mergers have a different structure than those launched in BNS mergers and rule out whether greater than or similar to 80% of binary neutron star mergers launch jets. We discuss the implications of future constraints on understanding the physics of short gamma-ray bursts and binary evolution.