If you come into contact with a randomly selected person who has been double-vaccinated, the chances of catching Covid off that person is much less than if you come into contact with someone who hasn't. The main and clearest reason is that the
people who have not been double-vaccinated are far more likely to be infected. Note that majority of the UK population HAS been double vaccinated, and yet the ABSOLUTE numbers of new infections in that group is significantly less, so the difference in probablity of those random people being infected is a lot greater than the obvious visual differences in the figure below.
Additionally, I believe those who test positive and are double vaccinated will have a less severe disease, and thus be less likely to pass the virus on than someone who has not been double-vaccinated.
I also have a sneaky feeling that there is a correlation between refusal to get vaccinated and refusal to wear a mask.