  The binary J0737-3039 A & B includes two pulsars in a highly relativistic
orbit. The pulsed radio flux from pulsar B brightens considerably during two
portions of each orbit. This phenomenon cannot be naturally triggered by the
illumination of $\gamma$-rays or X-rays from pulsar A or the bow shock around
pulsar B. Instead, we explain these periodic flares quantitatively as episodes
during which pairs from pulsar A's wind flow into the open field line region of
pulsar B and emit curvature radiation at radio frequencies within an altitude
of $\sim 10^8$ cm. The radio photons then travel through B's magnetosphere and
eventually reach the observer on the other side of the pulsar. Our model
requires that A's wind be anisotropic and that B's spin axis be somewhat
misaligned relative to the orbital angular momentum. We estimate the expected
$\gamma$-ray and X-ray emission from the system.
