One dose of mRNA vaccine will provide some protection against severe COVID-19 disease, while two doses will provide ~95% protection from severe COVID-19 disease. However, it does take time for the body to build immunity after vaccination. Someone could be infected with the virus just before or just after vaccination and get sick, because the vaccine didn’t have enough time to provide protection.

It is still possible to transmit COVID-19 to others after you have been vaccinated, though real-world data shows a significant reduction in transmission after the second dose with the mRNA vaccine. This is a significant reduction in transmission risk, but you should continue to follow the public health measures (masking, physical distancing, limits to social gatherings, etc.) recommended by the jurisdiction you live in.