Under what circumstances am I responsible for an ex's credit card debt if the card was shared jointly?

Pretty much any circumstance. If you are both listed as co-owners on the card, then you are both equally responsible for any, and all, charges that are made to the card, regardless of which one of you is actually doing the charging.
If she runs up the bill and doesn't pay, it will affect both of your credit scores and the card company can seek payment from both of you.
If you are both listed as co-owners, I would recommend closing the card/account by any means necessary so that you are not linked to her in that way. It is really the only way to terminate any responsibility you would have for charges that either of you made to the card.
If you are simply listed as an authorized user, I would ask her to have you removed in that capacity. Even as an authorized user, while you should not be financially responsible for the debts on the card (i.e. you should not have the responsibility to pay any charges, even if your ex does not) the card will show on your credit report and any derogatory marks, late payments, balances, collections associated with the card, etc. would appear as well. So, while you would theoretically have no financial responsibility if you are merely an authorized user, your credit could still be impacted regardless.
Hope this helps and good luck!