When bills start coming in after a death, the executor may worry about being personally on the hook for the deceased person's debts. But if that's your situation, you can relax. In most situations, you're not going to have to use your own money to pay estate debts.
Debts—ones the deceased person incurred while alive, or expenses the estate has after the death—should be paid for with estate property. For example, if the deceased person left a checking or savings account, the executor should transfer those funds into an estate bank account and use the money to pay bills.
If there isn't any cash immediately available—and bills have to be paid immediately—then the executor or family members may need to go ahead and pay some bills from their own funds. For example, you might need to pay burial or cremation expenses right away. Keep careful track of how much you spend, what you spend it for, and when you spend it. As long as you have good records, and the expense was necessary, you can reimburse yourself from estate funds later.
If you suspect that there aren't enough estate funds to pay the deceased person's debts, be careful about paying for anything, with either your own money or estate funds. Remember that under state law, creditors will have months (up to a year in some states) to come forward with their claims. State law will set out the priority in which debts should be paid; creditors at the bottom of the list will simply be out of luck. And if you give estate money to creditors who aren't entitled to receive it, you might have to reimburse the estate for the loss you caused it. (See below.)
In some circumstances, survivors may be personally liable for debts of the deceased person.
Surviving spouses are liable for debts the couple incurred together. For example, the survivor will be responsible for charges on a joint credit card, no matter which spouse actually charged the purchase.
If the deceased spouse incurred a debt alone, though, the survivor may not be liable. It depends on state law, the nature of the debt, and how the couple owned property together. For example, creditors owed money by only one spouse could probably go after the deceased spouse's half of property the couple owned together. But if the couple owned property "as tenants by the entireties," (allowed only in some states), then creditors could not reach that property for payment of the debts of just one spouse.
If someone cosigned for a loan or line of credit issued to the deceased person, the cosigner will be liable for the debt if the assets of the deceased person don't cover it. That's what cosigning is—promising to make good on a debt if the primary borrower, for whatever reason, cannot.
If the executor is careless or dishonest while in charge of estate assets, and the estate loses money as a result, the executor may be on the hook for certain debts. For example, say the executor, without waiting to add up the estate's debts and assets, quickly pays a large credit card bill of the deceased person. Later, it turns out that the estate doesn't have enough money to pay all of its debts—and some unpaid bills, including expenses of the last illness, have higher priority under state law than does credit card debt. The executor will probably have to reimburse the estate for the amount of money paid to the credit card issuer.