Although vaccinated people can be infected and even spread the virus, the disease is usually mild, amount of infectious virus and the time they are infectious is much smaller. And that is no speciality of covid immunizations, other vaccines like measles or influenza also don't prevent infection, but aim to prevent the disease.
The "not fully tested" meme is nonsense, many hundred million people have been vaccinated in the meantime, the safety profile is known.
Regarding long time effects beside immunity of vaccines, this interview https://lexfridman.com/vincent-racaniello/ goes into some detail. TLDL: there is nothing to expect/fear.