Thanks to Gaffi I could verify the hypothesis: As long as a user hasn't flagged any questions or answers, the flag counters are not shown at all.
Now that I have one flagged answer ("1 moderator attention flags, 1 waiting for review"), the flag counters, including comment flag counters, are shown on my flagging summary page.
And another observation: Comments you have flagged don't show the flag icon (on hover) while they are pending for review. If the flag icon comes back, the comment flag has been declined.