Such jurors are supposed to recuse themselves and be replaced by the already empaneled alternates.
Should such a jury wait until the end of the trial and then politely tell the judge that something has come up requiring his recusal, or will the judge ask if anything has come up, or how does that work? Certainly it would not be good for the juror to say, within earshot of other jurors, "Your honor, I don't think I could fairly judge this guy because I caught his character witness stealing from my store," but private communications between a juror and the judge would appear unseemly.