  Merging binary neutron stars are thought to be formed predominantly via
isolated binary evolution. In this standard formation scenario, the first-born
neutron star goes through a recycling process and might be rapidly spinning
during the final inspiral, whereas the second-born star is expected to have
effectively zero spin at merger. Based on this feature, we propose a new
framework for the astrophysical characterization of binary neutron stars
observed from their gravitational wave emission. We further propose a prior for
the dimensionless spins of recycled neutron stars, given by a gamma
distribution with a shape parameter of 2 and a scale parameter of 0.012,
extrapolated from radio pulsar observations of Galactic binary neutron stars.
Interpreting GW170817 and GW190425 in the context of the standard formation
scenario and adopting the gamma-distribution prior, we find positive support
(with a Bayes factor of 6, over the nonspinning hypothesis) for a spinning
recycled neutron star in GW190425, whereas the spin of the recycled neutron
star in GW170817 is small and consistent with our prior. We measure the masses
of the recycled (slow) neutron stars in GW170817 and GW190425 to be
$1.34_{-0.09}^{+0.12}$ $(1.38_{-0.11}^{+0.11})$ $M_{\odot}$ and
$1.64_{-0.11}^{+0.13}$ $(1.66_{-0.12}^{+0.12})$ $M_{\odot}$, with $68\%$
credibility, respectively. We discuss implications for the astrophysical
origins of these two events and outline future prospects of studying binary
neutron stars using our framework.
