Who is legally responsible for deceased person's debts?

As long as children do not co-sign or guarantee the credit card debt they are not responsible for estate debt.  When a person dies, their estate is liable for all of that person's debts.  As far as the priorities of debt please see my article entitled Pennsylvania Probate: What Debts Take Priority at the following link: http://www.sjfpc.com/PA_Probate_Law_Section_3392_Estate_Debt_Priorities.html

Although it relates to PA law most states have similar rules and statutes.

As a general rule, children do not inherit their parent's debt. That is, unless they specifically assumed any specific liability such debt (for example, they were a co-signer on a loan); a child bears no legal responsibility for a deceased parent's finances. The only time a child may possibly be held responsible for a parental debt is under the doctrine of something called "filial responsibility". Many states with these laws, require adult children to care for their parents financially. However when enforced (and they rarely are) it is only for reimbursement to Medicaid for nursing home expenses. 

Note:  Although a child bears no responsibility for parental debt, their parent's estate does bear such a responsibility. Accordingly, to the extent there are estate assets, creditors will need to be paid before any distribution is made to beneficiaries/heirs. If there are not enough assets to cover the outstanding debts, then those debts then the creditors right them off.