Not necessarily. COVID might be so transmissible that the entire world has been exposed to it enough to trigger an immune response but only establishes itself and becomes an infection in those who get a large enough viral load over a short enough time period.
It wouldn't produce an immunity, it would just keep the innate immune response active enough to prevent the average small cluster of COVID or flu viruses from replicating and growing into a full blown infection. This wouldn't even give the adaptive immune system a chance to develop an immunity to either of them.
If someone went to the COVID or flu ward and had someone cough in their face, they'd still get infected because that kind of exposure is generally too much for the innate immune system to catch.