Vaccines delivered to the arm deltoid muscle never promised mucosal immunity.
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7733922/
> The mucosal immune system is the largest component of the entire immune system, having evolved to provide protection at the main sites of infectious threat: the mucosae. As SARS-CoV-2 initially infects the upper respiratory tract, its first interactions with the immune system must occur predominantly at the respiratory mucosal surfaces, during both inductive and effector phases of the response. However, almost all studies of the immune response in COVID-19 have focused exclusively on serum antibodies and systemic cell-mediated immunity including innate responses.
That's why vaccinated people can still get infected (upper respiratory tract) and have no symptoms (serum antibodies). They can infect others (upper respiratory tract) when they are infected but don't have symptoms or a temperature check saying they should stay at home. Hence the CDC now recommends tests for vaccinated people who are exposed to Covid.
It was stated by Fauci and others in 2020 that current Covid vaccines are non-sterilizing. There are new Covid vaccines in development which provide sterilizing immunity, delivered via nasal inhaler instead of deltoid arm muscle. The MMR (measles, mumps, rubella) vaccine is sterilizing, i.e. prevents infection and transmission.