Estate Law Questions? Ask an Estate Lawyer.
Any debt he has would become his estate's debt, and the executor of his estate has the obligation to pay any outstanding debts, expenses, and taxes of the estate before making any distribution to the heirs of the estate. If he is your ex-husband, the debt collectors should not harass you about it--if they do, tell them to take a hike.
Any life insurance policy that has beneficiaries designated on the policy (and not payable to the individual's estate) are considered nonprobate assets and do not become part of the estate. When the insured passes away, just supply whatever info the insurance company requires (e.g., death certificate), and they should send checks out to the beneficiaries listed on the policy. If you are a beneficiary, that would go directly to you, and those funds would not get tied up in the probate court or be inventoried with any assets of his estate.
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His heirs should not be responsible for his debt; however, if they stood to inherit anything, it is possible that whatever assets/funds are in the estate could be eaten up by the debts instead of being distributed to the heirs.
If he does not have a will, it would be preferable for him to execute one. That allows him to decide who gets what and appoint an executor for his estate, instead of having the state government say what happens. Without a will, it usually falls to the surviving family members to decide who would petition the court to be the administrator of the estate and take care of everything.