Despite the media and government messaging that isn't how it works. If you've recently had the vaccine it will generally shorten the period you are infectious for before symptoms and if you've had the vaccine at all with the last few months it will generally shorten the period you are infectious for but the peak infectivity is still similar. It gets more complicated again because the most transmission events happen around the peak infectivity in the hours before someone becomes symptomatic and up to about 3 days later, which largely still happen vaccinated or not, while being vaccinated might mean you stop being infectious a few days earlier than someone unvaccinated that is towards the end of the infectious period and outside the window where the bulk of transmission events occur.
With some variants of Omicron the peak infectivity can be pushed a bit further out to 3-6 days where the vaccine has been seen to have the effect of quite quickly clearing the virus immediately after the peak which has a slightly bigger impact on the window where most transmission events occur but not a fundamental impact.
(With whatever variant, I suspect XBB.1.5 but can only guess, we had go around here though people were coming out with symptoms only 2-3 days after they must have been exposed to it - which means transmission was pretty early on!).
I'm generalising a bit there because it actually gets a lot more complicated than that and I don't fully understand all the mechanics involved.
The vaccines are robust at preventing hospitalisation and death in the more vulnerable which is backed up here
https://www.gov.uk/government/publications/covid-19-vaccine-weekly-surveillance-reports and overall do have some impact on transmission and prevalence in society but have a far more limited impact on direct transmission than has been made out - to the point the official studies persistently just wave it away with vague comments and "data not yet available" LOL.