  Using a Milky Way double neutron star (DNS) merger rate of 210 Myr$^{-1}$, as
derived by the Laser Interferometer Gravitational-Wave Observatory (LIGO), we
demonstrate that the Laser Interferometer Space Antenna (LISA) will detect on
average 240 (330) DNSs within the Milky Way for a 4-year (8-year) mission with
a signal-to-noise ratio greater than 7. Even adopting a more pessimistic rate
of 42 Myr$^{-1}$, as derived by the population of Galactic DNSs, we find a
significant detection of 46 (65) Milky Way DNSs. These DNSs can be leveraged to
constrain formation scenarios. In particular, traditional NS-discovery methods
using radio telescopes are unable to detect DNSs with $P_{\rm orb}$ $\lesssim$1
hour (merger times $\lesssim$10 Myr). If a fast-merging channel exists that
forms DNSs at these short orbital periods, LISA affords, perhaps, the only
opportunity to observationally characterize these systems; we show that toy
models for possible formation scenarios leave unique imprints on DNS orbital
eccentricities, which may be measured by LISA for values as small as
$\sim$10$^{-2}$.
