I don’t experience any side effects from either the first or second dose of covid19 vaccine. Does it Mean my Immune System Isn’t Working?

No.  Pretty much everyone where I work was vaccinated.  Other than a sore arm, many of us had no side effects.  The worst I heard was a woman who hit a 103 degree temperature.  My wife had zero side effects.  IT's like auto accidents, no one cares about the millions of cars that didn't wreck, just the few that did.  No one cares about the majority of people who no or minimal side effects, just those who did. 

You are fine.