mom in nursing hom with large debts

This is simple. Send them a cease and desist IN WRITING via certified mail, return receipt requested. Tell them to never contact you by any means ever again.

How to deal with them depends upon your legal status.
Debt collectors are prohibited from communicating with anyone other than the consumer in any manner that even intimates that they are communicating with regard to a debt. FDCPA 805(b). They may only call a third-party to obtain "locator information" pertaining to the consumer (FDCPA 804), and are strictly prohibited even in such calls to disclose the presence of any debt owed by the consumer.

So it depends on who is and who is not considered the "consumer."
FDCPA 805(d) defines a "consumer" for the purposes of debt collection as including "the consumer's spouse, parent (if the consumer is a minor), guardian, exector, or administrator."

Are you a legal guardian of your mother? If not, all communications with you regarding the debt are illegal. Only the "consumer," can send them a cease and desist letter under FDCPA 805(c). They already have a statutory cease and desist requirement if you are not the consumer.

If you are not the legal guardian, simply tell them that their communications with you are in violation of FDCPA 805(b), and any furthr communication with you will result in complaint to the FTC or possible civil action for violation of the FDCPA.
If you are the legal guardian, then you can send a cease communications letter under section 805(c).