1. They reduce the chances you will get it because they increase the chances your system will fight it off before it gets established.
2. They reduce the time during which you are infectious, and the amount of the virus in your system that you output during that time. As a result they decreases the chances of you infecting others.
The reduction for the original variants was very significant. Vaccinated people were very unlikely to infect anyone else prior to Delta. With Delta the benefits were significantly reduced. With Omicron the benefits in infectiousness are again much reduced.
So for most of the last year the vaccines were a vital tool in reducing the spread of the virus as well as protecting the vaccinated. Now Omicron is so infectious the effect is much lower. Of course the vaccines do still seem to be real life savers against Omicron, and avoiding getting sick yourself still protects others by reducing the load on the health care system.
Transmission potential of vaccinated and unvaccinated persons infected with the SARS-CoV-2 Delta variant in a federal prison, July—August 2021 https://www.medrxiv.org/content/10.1101/2021.11.12.21265796v...
> A total of 978 specimens were provided by 95 participants, of whom 78 (82%) were fully vaccinated and 17 (18%) were not fully vaccinated. No significant differences were detected in duration of RT-PCR positivity among fully vaccinated participants (median: 13 days) versus those not fully vaccinated (median: 13 days; p=0.50), or in duration of culture positivity (medians: 5 days and 5 days; p=0.29).
https://www.imperial.ac.uk/news/231557/covid-vaccines-effect...
I’m not entirely surprised there are studies with small numbers of participants in which it doesn’t show up.
Your conclusion is essentially the right one, the leading vaccines (particularly the mRNA vaccines) can dramatically lower the risk of a bad outcome if you catch SARS2. They do not stop the spead of the virus and do not ensure you won't catch it.