Can a joint checking account holder be responsible for debts owed by the other party?

I am so sorry for your loss.  First, the debts of the decedent are debts of their estate and not any other person as an individual.  I think that you may need to have a clearer understanding of how joint accounts work.  Certain bank accounts and certificates may be owned with rights of survivorship, which means that upon the death of one party to the account, the surviving party (or parties) become the sole owner (owners). If the decedent maintained such an account, the survivor will be able to withdraw one-half of the funds in the account by giving the bank a death Certificate and without the need to provide anything from the Surrogate. The other half will not be released until a tax waiver is issued by the New Jersey transfer Inheritance Tax Bureau, normally after the tax is paid and the return is filed.

Under the new inheritance tax laws governing estates from spouse to spouse, when the bank account is co-owned, funds may be transferred with a copy of the death certificate, without any type of certificate from the surrogate's office. The spouse will sign an L-8 tax waiver, usually completed by the financial institute. If the account is in the name of the decedent only, the bank will require a certificate from the surrogate in addition to the L-8.

When the bank account is co-owned by any other Class A, (parents, grandparents, children, grandchildren, adopted children, or stepchildren), the procedure is the same as spouse to spouse, except the co-owner will sign an L-8 tax waiver. 

You have not indicated here how the parties are related.  I hope this answer gave some guidance.  Good luck.