A person tested negative has just not been proven positive, which is different from being proven negative. PCR test sensitivity can be as low as 70%. https://doi.org/10.1101/2020.04.16.20066787
A vaccinated person may only spread if they are indeed infected, which is less common to begin with. Then, if they are indeed infected, the rate of transmission is much lower - as they usually don't develop heavy viral loads.