Only the assets of the estate are used to pay debts incurred by the decedent. Beneficiaries and the executor are not personally liable for such debts (unless they are co-debtors or have personally guaranteed the debts of the decedent in some way).
Typically, once all assets are liquidated and that balance is used paid to the existing debtors, the remaining balance is simply written off by the debtors (unless they believe additional assets were available within the estate in some way).