Most gravitational-wave signals from binary neutron star coalescences are too weak to be individually resolved with current detectors. We demonstrate how to extract a population of subthreshold binary neutron star signals using Bayesian parameter estimation. Assuming a merger rate of one signal every 2 hours, we find that this gravitational-wave background can be detected after approximately 3 months of observation with Advanced LIGO and Virgo at design sensitivity, versus several years using the standard cross-correlation algorithm. We show that the algorithm can distinguish different neutron star equations of state using roughly 7 months of Advanced LIGO and Virgo design-sensitivity data. This is in contrast to the standard cross-correlation method, which cannot.