That person's 'estate' would pay all debts .. Which means any cash that person has in their bank account, money collected from anything that that person owns would be sold or even insurance money collected would pay for that person's debts. If there is nothing to to satisfy those debts, then the debtholder would be forced to take the loss.
If a person has a charge card with only his or her name on it and that person dies, who pays the balance?
If there is any estate left after the person dies, they will collect from that. Otherwise, no one else is responsible for the charges and the card company has to write it off.