While getting vaccinated is a massively good idea both for the individual and society it does appear to be the case that people who are vaccinated or were diseased and recovered and have antibodies that way that are exposed to delta virus can nonetheless develop high viral loads, high enough to be contagious to people who are not immune. It does appear that in vaccinated or otherwise immune people the viral load drops very quickly.

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.