  The conventional wisdom, dating back to 2012, is that the mass distribution
of Galactic double neutron stars is well-fit by a Gaussian distribution with a
mean of $1.33 M_\odot$ and a width of $0.09 M_\odot$. With the recent discovery
of new Galactic double neutron stars and GW170817, the first neutron star
merger event to be observed with gravitational waves, it is timely to revisit
this model. In order to constrain the mass distribution of double neutron
stars, we perform Bayesian inference using a sample of 17 Galactic double
neutron stars effectively doubling the sample used in previous studies. We
expand the space of models so that the recycled neutron star need not be drawn
from the same distribution as the non-recycled companion. Moreover, we consider
different functional forms including uniform, single-Gaussian, and two-Gaussian
distributions. While there is insufficient data to draw firm conclusions, we
find positive support (a Bayes factor of 9) for the hypothesis that recycled
and non-recycled neutron stars have distinct mass distributions. The most
probable model---preferred with a Bayes factor of 29 over the conventional
model---is one in which the recycled neutron star mass is distributed according
to a two-Gaussian distribution and the non-recycled neutron star mass is
distributed uniformly. We show that precise component mass measurements of
$\approx 20$ double neutron stars are required in order to determine with high
confidence (a Bayes factor of 150) if recycled and non-recycled neutron stars
come from a common distribution. Approximately $60$ are needed in order to
establish the detailed shape of the distributions.
