The vaccine hasn’t been proven to prevent the spread of the virus. It has only been proven to prevent the vaccinated person from becoming ill.
I’m not a doctor, but the way I understand it, a vaccine doesn’t prevent a virus from entering your body, it teaches your body how to fight the virus so that you don’t get sick (and that’s what the current vaccines have been proven to do). That doesn’t necessarily mean you can’t transmit the virus to others.