Binary pulsars most likely did not form from stars that started only millions of miles apart. They probably did spiral closer together during the super giant phases. When the first one exploded, the second one certainly lost some outer material, but a quick calculation shows that only a minor (and inert) amount of the mass of the second one would be lost in the explosion.
When the second star becomes a supergiant, the first neutron star should be able to orbit through the material of the other star like a truck driving through fog. The magnetic field of the first neutron star would have an impact on the supergiant though. Eventually, the second star explodes, and the energy transfered to the first neutron star is minimal compared to the forces holding it together, and keeping it in an orbit.
As far as I know this pair is the only known case where both neutron stars appear to us as pulsars. There are other known cases of pulsars orbiting neutron stars.
This pair looks like they will join in something like 85 million years if I remember correctly.
Forming opinions as we speak