Medicaid is a means-tested public benefit that is funded by both the federal and state governments. So yes, the joint-sponsor would be liable for the cost of this benefit if used (even in an emergency basis) by the immigrant.
All the sponsor's income is taken into consideration for meeting the eligibility requirements, but that income must be proven.
And no, the sponsor is not responsible for means-tested public benefit that the immigrant has used in the past before the sponsor signs the Affidavit of Support.
Sorry for the delay in getting back to you. Since the I-864P specifies that emergency Medicaid (though not cash emergency care) is not included, then the joint sponsor would not be responsible for such a benefit.
By the way, one method that the joint sponsor gets out of the responsibility is if the immigrant becomes a U.S. Citizen.
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