In general, if you are the only person legally obliged to pay the debt before your death, then your probate estate becomes responsible for paying it after your death. If all of your debts are paid from your probate estate, then no one else needs to worry about them. If your probate assets are not sufficient to pay all of the administrative expenses and debts you have, then for the most part, the unpaid debts just have to be written off and they do not become someone else's responsibility. However, if an heir or a beneficiary receives assets from the probate estate and there are still unpaid debts, the unpaid creditors can go after the person who received the assets for the value of the assets. That is because beneficiaries or heirs are last in line to receive your probate assets; creditors come before them.
When you die, someone (an Executor, if you have a Will, or an Administrator, if you don't have a Will) has to go through your assets to determine what becomes part of your probate estate. Certain assets may be transferred at your death under a beneficiary designation (like life insurance or an IRA usually do), or by a right of survivorship (typically, joint bank or brokerage accounts have a right of survivorship attached to them; certain, but not all, jointly held real estate can also have a right of survivorship attached to it). If an asset is transferred directly to an individual (not to your estate) at your death by a beneficiary designation or a right of survivorship, that asset does not become part of your probate estate and MAY not be subject to creditor claims (many, but not all, assets that pass by a beneficiary designation or a right of survivorship are protected from your creditors at your death). The Executor or Administrator is then responsible for figuring out what debts you have and ensuring that they get paid in the proper order (some debt has a higher status than other debt, and if they aren't paid in the right order and some debt remains unpaid, the Executor or Administrator can end up becoming personally responsible for some of the unpaid debts). This is part of the estate administration process (note: that's actually NOT part of the probate process; it's related but separate).
Whether you have a Will or not does not affect the debt question, although it is generally a good idea to have one because it makes the process of getting someone appointed to handle your estate much easier and makes that person's job easier, too (if the Will is well-drafted and correctly executed, at least).
That said: if there is a co-borrower on the debt, that person continues to also be responsbile for that debt, and if the debt is a secured debt like a mortgage or car loan, then the security interest still exists and can be foreclosed on if the debt is not paid....
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