A unified strategy is developed that can be used to search for millisecond pulsars (MSPs) with ~solar mass companions (including neutron star companions in double neutron star binaries [DNSBs]) belonging to both very short period binaries, and those with periods so long that they could be appropriate targets for acceleration searches, and to bridge the gap between these two extremes. In all cases, the orbits are assumed to be circular. Applications to searches for binary pulsars similar to PSR J0737-3039 are discussed. The most likely candidates for more DNSBs consist of weakly magnetized neutron stars, spinning only moderately fast, like J0737-3039A, with periods generally longer than 15 ms, though this issue is not yet settled. Because of the similarity between the MSP components of DNSBs, and the longer period MSP population specific to massive condensed or core collapsed globular clusters, as well as the uncertainties about accretion-driven spinup, doubts linger about the standard models of DNSB formation.