Specially trained agents are calling the heirs of the recently departed and politely asking them to pay up the outstanding balances on credit card debt, car loans, and the like. If an estate is filing an official probate proceeding, creditors have a process for filing claims against the decedent's property. But that's not who these collection agents are calling.
They're calling the relatives of people who have died who aren't in formal probate proceedings either because they had created living trusts during their lifetime or because the estate was too small for a probate proceeding.
Here's my advice: No matter how nice these people are (or well-trained), simply ask them why they think that you're legally responsible for the debt. In the vast majority of cases, you aren't obligated to pay them a nickel. It's true that married people can be jointly liable for debts, and creditors can try and get repaid from the property you've inherited from the decedent depending on your state's laws -- but that's their problem, not yours.
In fact, most estate planning attorneys can successfully get credit card companies and the like to accept a fraction of the outstanding balance on those debts that the survivors are responsible for, because these companies know just how hard it really is to collect anything on the dead's debts.
For a comprehensive guide to the legal, personal and practical aspects of administering a living trust, see The Trustee's Legal Companion, by Liza Hanks and Carol Elias Zolla (Nolo).