I think this is from a misunderstanding of what transmissibility people were talking about. People getting the vaccine did not transmit the disease to others simply by virtue of being vaccinated. What that means is, you can get the shot and not worry that, by getting the shot, you are going to transmit the disease to others.
What that does not mean is that, once vaccinated, you cannot ever transmit the disease to others. If you get sick, the vaccine will likely make you get over it faster and have it less impactful, reducing transmission as an added bonus of those two things happening (you're coughing less so less is put in the air, you are sick for less time so the period you will be heavily contagious for and also symptomatic is shorter) but it will not entirely prevent transmission.