We investigate further the dynamics of colliding winds in hot-star binary systems, by concentrating in this paper on the effects of the radiation pressure of both stars on the material moving between them. Naturally, the presence of a luminous binary companion tends to inhibit the wind acceleration towards the stagnation-point because the resultant radiation force is smaller than in a single star. By also considering the countervailing influence of the companion's gravitational attraction, we derive inequalities governing the stellar mass-loss flux. We find that, in close binary systems, such as V444 Cyg, the winds collide at much lower head-on velocities than those implied by the single-star models used in earlier calculations. This leads directly to lower expected X-ray luminosities and temperatures for colliding hot-star winds, and we suggest that this accounts for the long-standing discrepancies between theory and observation.