There’s a study out demonstrating that vaccinated people with breakthrough infections are significantly less likely to transmit the virus than infected unvaccinated people, even when viral loads are identical.
The study followed 24,706 healthcare workers and observed 161 infections, combining data for both AZ and Pfizer vaccines. The viral loads as measured by PCR amplification were lower among the vaccinated, but not significantly so (P = 0.53):
But then they actually tried to culture the virus from the samples:
An identical viral load is not as infective in a vaccinated person. The reason for this is explained here:
https://twitter.com/SmallRedOne/status/1425133972523257863
The averages were 84.9% infectious for unvaccinated and 68.6% for vaccinated (P = 0.005) so being vaccinated makes 1 in 6 COVID-positive people not infectious when they otherwise would have been - obviously on top of making it much less likely to get infected in the first place.