Using a Milky Way (MW) 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 MW for a 4 yr (8 yr) 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) MW DNSs. These DNSs can be leveraged to constrain formation scenarios. In particular, without prior information on a particular system's position and orbital period, traditional NS-discovery methods using radio telescopes alone are insensitive to DNSs with P-orb less than or similar to 1 hr (merger times less than or similar to 10 Myr). If a fast-merging channel exists that forms DNSs at these short orbital periods, LISA affords, perhaps, the best opportunity to observationally identify and characterize these systems; we show that toy models for possible formation scenarios leave imprints on DNS orbital eccentricities, which may be measured by LISA for values as small as similar to 10(-2).