The winds of massive stars are important for their direct impact on the interstellar medium, and for their influence on the final state of a star prior to it exploding as a supernova. However, the dynamics of these winds is understood primarily via their illumination from a single central source. The Doppler shift seen in resonance lines is a useful tool for inferring these dynamics, but the mapping from that Doppler shift to the radial distance from the source is ambiguous. Binary systems can reduce this ambiguity by providing a second light source at a known radius in the wind, seen from orbitally modulated directions. From the nature of the collision between the winds, a massive companion also provides unique additional information about wind momentum fluxes. Since massive stars are strong ultraviolet (UV) sources, and UV resonance line opacity in the wind is strong, UV instruments with a high resolution spectroscopic capability are essential for extracting this dynamical information. Polarimetric capability also helps to further resolve ambiguities in aspects of the wind geometry that are not axisymmetric about the line of sight, because of its unique access to scattering direction information. We review how the proposed MIDEX-scale mission Polstar can use UV spectropolarimetric observations to critically constrain the physics of colliding winds, and hence radiatively-driven winds in general. We propose a sample of 20 binary targets, capitalizing on this unique combination of illumination by companion starlight, and collision with a companion wind, to probe wind attributes over a range in wind strengths. Of particular interest is the hypothesis that the radial distribution of the wind acceleration is altered significantly, when the radiative transfer within the winds becomes optically thick to resonance scattering in multiple overlapping UV lines.