We report here the first polarimetric measurements of the pulsars in the J0737 - 3039 binary neutron star system using the Green Bank Telescope. The data suggest that the primary star ( pulsar A) has a wide hollow cone of emission, which is an expected characteristic of the relatively open magnetosphere given its short spin period, and that pulsar A has a small angle between its spin and magnetic dipole axes,. This geometry suggests 4degrees +/- 3degrees. that pulsar A's wind pressure on pulsar B's magnetosphere will depend on orbital phase. This variable pressure is one mechanism for the variation of flux and profile shape of pulsar B with respect to the orbital phase that has been reported. The response of pulsar B to the pulsar A wind pressure will also depend on the particular side of its magnetosphere facing the wind at the spin phase when pulsar B is visible. This is a second possible mechanism for variability. We suggest that pulsar B may have its spin axis aligned with the orbital angular momentum owing to pulsar A's wind torque that contributes to its spin- down. Monitoring the pulsars while geodetic precession changes spin orientations will provide essential evidence to test detailed theoretical models. We determine the rotation measures of the two stars to be -112.3 +/- 1.5 and -118 +/- 12 rad m(-2).