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)