  We discuss the effects of particle acceleration at the bow shocks expected in
the binary pulsar system J0737-3039, due to the wind from pulsar A interacting
with both the interstellar medium (ISM), and with the magnetosphere of pulsar
B. In this model, we find that the likeliest source for the X-rays observed by
Chandra is the emission from the shocked wind of pulsar A as it interacts with
the ISM. In this case, for favorable model parameter values, better statistics
might help Chandra marginally resolve the source. A consequence of the model is
a power law high energy spectrum extending up to $\lesssim$ 60 keV, at a level
of $\sim 2 \times 10^{-13}$ erg cm$^{-2}$ s$^{-1}$.
