If I co-sponsor an immigrant am I responsible for any debt they incur?

As a joint sponsor, you are agreeing to use your resources to support your son-in-law, if it becomes necessary. If your son-in-law receives a Federal, State, or local means-tested public benefit, the agency providing the benefit may request that you repay the cost of those benefits. These benefits generally include Food Stamps, Medicaid, Supplemental Security Income (SSI), or Temporary Assistance for Needy Families (TANF). For immigration purposes, being a joint sponsor, in and of itself, would not make you liable for debts your son-in-law incurs.

No, but if he receives any means-test public benefits, you will be responsible.

No, as my colleagues pointed out quite well, you will NOT be responsible for 'any' debt he incurs. He's on his own with car loans, credit card bills, etc. But, if he tries to collect any 'welfare-type' benefits, he needs to show your income ... which will probably 'knock' him out of being able to get those benefits.

No. Not at all. This link explains the affidavit in California and why it isn't such a big deal as it is in some other states: http://www.nilc.org/sponsoredimms&bens-ca.html