While they can, studies have shown that it is far less likely. One of the most illustrative examples of this deals with school-age kids and teachers, where virus-positive kids with no symptoms are not spreading it to teachers. The studies support this, and even on the CDC site, they basically mention that asymptomatic could happen in passing, rather than having a full blown discussion like they have about the major vectors of transmission.
"Not fully sick" means the person is not asymptomatic, but rather it means they are sick, and can spread the virus in a major way, especially if they are coughing/sneezing.