  We revisit the formation of PSR J0737-3039, taking into account the most
recent observational constraints. We show that the most likely kick velocity
and progenitor parameters depend strongly on the consideration of the full
five-dimensional PDF for the magnitude and direction of the kick velocity
imparted to pulsar B at birth, the mass of pulsar B's pre-supernova helium star
progenitor, and the pre-supernova orbital separation, and on the adopted prior
assumptions. The priors consist of the transverse systemic velocity, the age of
the system, and the treatment of the unknown radial velocity. Since the latter
cannot be determined from observation, we adopt a statistical approach and use
theoretical radial-velocity distributions obtained from population synthesis
calculations for coalescing double neutron stars. We find that the prior
assumptions about the pre-supernova helium star mass affect the derived most
likely parameters significantly: when the minimum helium star mass required for
neutron star formation is assumed to be 2.1Msun, the most likely kick velocity
ranges from 70-180km/s; when masses lower than 2.1Msun are assumed to allow
neutron star formation, the most likely kick velocity can be as low as a few
km/s, although the majority of the considered models still yield most likely
kick velocities of 50-170km/s. We also show that the proximity of the double
pulsar to the Galactic plane and the small proper motion do not pose stringent
constraints on the kick velocity and progenitor mass of pulsar B. Instead, the
constraints imposed by the orbital dynamics of asymmetric supernova explosions
turn out to be much more restrictive. We conclude that the currently available
observational constraints cannot be used to favor a specific core-collapse and
neutron star formation mechanism. (abridged)
