Signing the Affidavit of Support, means you accept legal responsibility for financially supporting the sponsored immigrant(s) until they become U.S. citizens or can be credited with 40 quarters of work. Any joint sponsors or household members whose income is used to meet the minimum income requirements are also legally responsible for financially supporting the sponsored immigrant.
Thus, if the immigrant receives any "means-tested public benefits," a sponsor/co-sponsor is responsible for repaying the cost of those benefits to the agency that provided them. Failure to repay the debt means the agency can sue you in court to get the money back.
Some examples of Federal means-tested public benefits include Food Stamps, Medicaid, Supplemental Security Income (SSI), Temporary Assistance for Needy Families (TANF), and the State Child Health Insurance Program (CHIP). States and local jurisdictions may also designate certain of their programs as means-tested public benefits. The following types of programs are not means-tested public benefits: emergency Medicaid; short-term, non-cash emergency relief; services provided under the National School Lunch and Child Nutrition Acts; immunizations and testing and treatment for communicable diseases; student assistance under the Higher Education Act and the Public Health Service Act; certain forms of foster-care or adoption assistance under the Social Security Act; Head Start programs; means-tested programs under the Elementary and Secondary Education Act; and Job Training Partnership Act programs.
If you have any questions regarding sponsoring an alien you should speak with an immigration attorney.