Isolation (or quarantine) post-exposure and testing for close contact of a suspected coronavirus infected patient is required to limit the spread of contagion, and seek help if anybody does develop symptoms or feels sick.
While there are no clear guidelines in place, some clinical guidelines believe that vaccinated individuals may have to follow relaxed norms, or, in some cases, not really have to fully isolate themselves. While isolation helps protect others from the risk of exposure, CDC guidelines specify that unless and until a vaccinated individual, who is exposed to someone with the illness develops symptoms himself, there is no real need to distance themselves from others. Wearing a mask, maintaining some distance, just as general precautions recommend would be sufficient.
However, testing would be recommended so as to cut down specified risks. Several vaccinated people may also happen to be asymptomatic, or develop fairly mild symptoms so getting tested would offer a level of assurance.