The recent LIGO-Virgo detection of gravitational waves from a binary neutron star merger event GW170817 and discoveries of its accompanying electromagnetic signals marked a new era for multi-messenger astronomy. In the coming years, advanced gravitational-wave detectors are likely to detect tens to hundreds of similar events. Neutron stars in binaries can possess significant spin, which is imprinted on the gravitational waveform via the effective spin parameter. We develop a fiducial model, informed by radio observations, in order to predict the effective spin of merging binaries like GW170817. We explore the astrophysical inferences made possible by gravitational-wave measurements of spin. First, using our fiducial model, we predict that around 15-40% of binary neutron stars should have non-negligible spins assuming the spin axis of the recycled neutron star aligns with total orbital angular momentum of the binary. Second, a population for which the recycled neutron star spinning at a few milliseconds periods can be distinguished from more typical systems after 10 detections. Third, with more than 50 detections, we can confidently tell whether or not the spin axis of the recycled neutron star tends to be aligned with the binary orbit. Finally, stringent constraints can be placed on neutron star magnetic field decay after 200 detections.