They can take the deduction. They have what is called "equitable" interest in the property.
"Treas. Reg. § 1.163-1(b)
permits a deduction for interest paid on a mortgage when a taxpayer is the
legal or equitable owner of the property, even though the taxpayer is not
directly liable for the mortgage."
A taxpayer becomes the beneficial or
equitable owner of mortgaged property (allowing him to deduct mortgage interest
on that property) when he assumes the benefits and burdens of ownership.
Whether, and when, this has occurred is determined under state law, on a
case-by-case basis.
In determining whether the benefits and burdens of ownership
have been transferred to a taxpayer, the Tax Court often considers whether the
taxpayer (1) has a right to possess the property and enjoy its use, rents, or
profits, (2) has a duty to maintain the property, (3) is responsible for
insuring the property, (4) bears the property's risk of loss, (5) is obligated
to pay the property's taxes, assessments, or charges, (6) has the right to
improve the property without the owner's consent, and (7) has the right to
obtain legal title at any time by paying the balance of the purchase price.
"
