When you settle debt, the creditor is required to send the Internal Revenue Service a notice of the amount of forgiven debt. Legally, when you co-sign on a debt, it belongs to you.
Fortunately, however, the Internal Revenue Service takes a logical view of forgiven debt for co-signers. Treasury Regulation Sec. 1.6050P-1(7) says that for purposes of filing, "a guarantor is not a debtor." It differentiates between those who get the benefit of a loan and those who guarantee its repayment. You are a guarantor, and so you should be off the hook.
As a co-signer who did not received the loan money or benefit from it, you should not receive a Form 1099-C from the lender. If you do receive Form 1099-C, you should contact the lender and tell them to correct the mistake. Do not include the forgiven debt on your tax return.
The buyer, however, should receive a Form 1099-C for the $10,000. Don't automatically assume the buyer's forgiven debt is taxable, however. There are several exceptions to the rule, including the insolvency rule. That means your the buyer does not have to pay tax on the forgiven debt to the extent that their liabilities exceeded their assets prior to the cancellation of debt.
To claim the insolvency exemption, the buyer should first use the worksheet in IRS Publication 4681, Canceled Debts, Foreclosures, Repossessions and Abandonments, to calculate their total assets and total liabilities.
Then, they should use the results from the worksheet to fill out IRS Form 982, Reduction of Tax Attributes Due to Discharge of Indebtedness, and attach it with the Form 1099-C to their tax return.
If the amount by which the buyer is insolvent (the amount her total liabilities exceed their total assets) is greater than the amount of forgiven debt, they don't have to pay tax on the "income" from forgiven debt.
If the forgiven debt is greater than the amount by which the buyer is insolvent, however, they must pay tax on a portion of the forgiven debt. For example, say when they fill out the worksheet in IRS Publication 4681 and determines that they is insolvent by $8,000. She has $10,000 in forgiven debt on Form 1099-C. They can exclude $8,000 from income, but they must report $2,000 as other income on their income tax return ($10,000 - 8,000 = $2,000).