Can vaccinated people still spread COVID?

I asked the same question of Google. The best article (that The NY Times let me read!) was at:
https://www.nytimes.com/2020/12/08/health/covid-vaccine-mask.html

The theory seems to be that antibodies produced as a result of the vaccine may have a hard time penetrating the mucous layer that surrounds the nose and other parts of the respiratory system. Since the Pfizer and Moderna vaccines are injected into a muscle, the thinking seems to be that not enough antibodies would make it into the nose. Therefore, even vaccinated people could carry enough virus in the nose to be a danger to others.

Personally I find the theory defective on two counts:

1. The human body is much smarter than doctors and scientists think.
2. The virus is too smart to just stay in the nose.

What do you think?