Generally, beneficiaries are not responsible for debts incurred by the deceased alone (as opposed to debts incurred jointly by the deceased and a beneficiary, such as where a beneficiary is the guarantor of the debt incurred by the deceased).  Under certain circumstances, however, a creditor may argue that the deceased made fraudulent transfers shortly before death in order to avoid her obligations to creditors.  For example, if I gave my house to my daughter two days before I died, knowing my death was imminent, a creditor might object to that transfer and seek to have the property returned to my estate.  In almost all other situations, the beneficiaries need not worry about the decedents debts, EVEN IF the creditors attempt to collect the debt from the survivors (as they sometimes do).

On the other hand, the ESTATE of the deceased is liable for her debts.  Any probate assets (things owned by the decedent at the time of her death) will be sold in order to pay the debts of the estate (the decedent).  In that case, beneficiaries who thought they might be in line to inherit some amount of money, may find that they get nothing or a reduced amount.  Follow this link to a post with more information about your obligations regarding probate assets and the probate process.