There are actually two parts to the question. The first is whether the assisted living facility itself has a legal right to bill you and your sister for your mom's
care. The answer is no. Adult children are not responsible for the bills or debts of their parent, even if they are now the legal owners of assets that used to belong to the parent.
The second part of the question is what happens if your mom cannot pay for long-term care and applies for Medcaid coverage – either for her assisted living facility, if it participates in Medicaid, or for a nursing home. In that case, in deciding eligibility Medicaid looks at any transfer of assets by your parent, for less than full value, during what's called a "look-back" period. For transactions before February 8, 2006, Medicaid considers any asset transferred within the previous 36 months to disqualify for Medicaid, for a period of time from the transfer date, the person who made the transfer; the length of the disqualification is based on the value of the asset and the cost of care in the state. Since your mom transferred her house to you and your sister more than 5 years ago, though, this rule wouldn't apply to you and Medicaid would not consider the house at all in determining your mom's Medicaid eligibility.