Great article on this:
TL;DR version: As others have said, outside of joint account relationships, the debt belongs to the estate, and others have no responsibility to pay. But the article details a whole industry, death-debt collectors, who target survivors who might agree to pay at least part of what the dead person owed.
Debt collectors say the survivors have a moral obligation to pay, especially in cases where they benefited from purchases rung up by someone else.
"Collectors are starting to realize just how much money you can get from someone when they are at their most vulnerable,"
Debt collectors often tell surviving family members that they aren't personally responsible for paying the debts of the deceased. But those words barely register with grieving relatives, according to interviews with a dozen lawyers who represent about 60 families pursued for money owed by dead relatives.
"Each call brought up fresh memories of my husband's death," Patricia Smith, 56, says about the calls she started getting last year about $1,787.04 in credit-card debt owed by her late husband, Arthur.
The debt-collection calls and letters kept coming and wore her down, says Mrs. Smith, who lives in Jackson, Miss. She agreed to scrounge together $50 a month "just to make the calls stop."
Note that this article faults various policies by the FTC, who reply strongly (but to my mind unconvincly) in the comments