Earlier quoted context omitted.
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vaccinated people may carry the virus, but they're a lot less infectious than un-vaccinated.
So it's not quite as simple as people with an axe to grind will claim it is, but it is possible for a vaccinated person to briefly have the same viral load as an unvaccinated person. The duration of that dangerous period is much shorter when vaccinated, and of course the risk to the vaccinated person is much, much lower.