  Recent pulsar surveys have increased the number of observed double neutron
stars (DNS) in our galaxy enough so that observable trends in their properties
are starting to emerge. In particular, it has been noted that the majority of
DNS have eccentricities less than 0.3, which are surprisingly low for binaries
that survive a supernova explosion that we believe imparts a significant kick
to the neutron star. To investigate this trend, we generate many different
theoretical distributions of DNS eccentricities using Monte Carlo population
synthesis methods. We determine which eccentricity distributions are most
consistent with the observed sample of DNS binaries. In agreement with
Chaurasia & Bailes (2005), assuming all double neutron stars are equally as
probable to be discovered as binary pulsars, we find that highly eccentric,
coalescing DNS are less likely to be observed because of their accelerated
orbital evolution due to gravitational wave emission and possible early
mergers. Based on our results for coalescing DNS, we also find that models with
vanishingly or moderately small kicks (sigma < about 50 km/s) are inconsistent
with the current observed sample of such DNS. We discuss the implications of
our conclusions for DNS merger rate estimates of interest to ground-based
gravitational-wave interferometers. We find that, although orbital evolution
due to gravitational radiation affects the eccentricity distribution of the
observed sample, the associated upwards correction factor to merger rate
estimates is rather small (typically 10-40%).
