The double pulsar system J0737-3039 appears associated with a continuous radio emission, nearly three times stronger than that of the two pulsars together. If such an emission comes from a tranparent cloud its spatial extent (> 10^13 cm) should be substantially larger than the orbital separation. Assuming homogeneity and equipartition, the cloud magnetic field is 0.03 G and the electron characteristic energy ~ 60 MeV. This is consistent with supposing that relativistic electrons produced in the shock formed by the interaction of the more luminous pulsar wind with the magnetosphere of the companion flow away filling a larger volume. Alternatively, the unpulsed emission may directly come from the bow shock if some kind of coherent mechanism is at work. Possible observational signatures that can dicriminate between the two pictures are shortly discussed.

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