We present a model for atmospheric wind circulation in binary millisecond pulsar (MSP) companions, showing how the optical light curve (LC) and radial velocities are sensitive to the wind flow, causing LC orbital phase shifts and asymmetries, as observed for several "spider" MSPs. Velocity widths of spectral lines offer additional opportunities for measuring surface wind speed. As examples, we fit optical data for the black widow pulsar J1959+2048 and the redback pulsar J2215+5135; the wind heating models (WH) are statistically strongly preferred over direct heating (DH) for both objects, although the latter is even better fit with a heated spot. In general, WH effects tend to increase the inferred orbital inclination I and decrease the inferred companion center-of-mass radial velocity amplitude Kc; both effects decrease the inferred neutron star mass. Even with such a decrease, we find large masses for the two neutron stars: 2.18 ± 0.09 M☉ and 2.28-0.09+0.10 M☉, respectively (for the modest surface speeds fit from the bulk heat flow; supersonic photospheric winds can slightly change these values). These are among the highest masses known, and our improved modeling increases confidence that the results are important for understanding the dense matter equation of state.