The co-signer is not responsible and should not receive a 1099-C. See this answer for more.
https://ttlc.intuit.com/community/taxes/discussion/do-both-borrower-cosigner-file-1099-c/00/1877982
You should contact the lender and ask them to cancel the 1099-C. If they refuse, you should list it as income on your tax return and then create an item of "other miscellaneous income" in a negative amount (minus sign) to offset the income, for the reason, say something like "1099-C not taxable to co-signer".
I can't answer whether the 1099-C is taxable to the student borrower. The rules on that have changed a lot and it depends on why the loan was canceled and who made the loan. Private loans may have different rules than government loans.
