Publication 936 is somewhat clear about this situation. (emphasis mine below)

If you and at least one other person (other than your spouse if you file a joint return) were liable for and paid interest on a mortgage that was for your home, and the other person received a Form 1098 showing the interest that was paid during the year, attach a statement to your return explaining this. **Show how much of the interest each of you paid,** and give the name and address of the person who received the form. Deduct your share of the interest on Schedule A (Form 1040), line 11, and print “See attached” next to the line. Also, deduct your share of any qualified mortgage insurance premiums on Schedule A (Form 1040), line 13.

Similarly, if you are the payer of record on a mortgage on which there are other borrowers entitled to a deduction for the interest shown on the Form 1098 you received, **deduct only your share of the interest** on Schedule A (Form 1040), line 10. Let each of the other borrowers know what his or her share is.

What matters is who paid how much. If you each paid half, you can each deduct half. If one person paid the whole thing, they can deduct the whole thing. There doesn't seem to be any freedom to allocate the deduction in any way other than based on actual payments.

However your situation could be interpreted two ways. The full amount came out of her checking account, so perhaps she is entitled to the full deduction. However you repaid her exactly half each month, so perhaps you're each considered to have paid half. I'm not sure if there's any official guidance on it. Seems to me like half-and-half is the more logical choice.

Others are right that you may find it more advantageous for one or the other of you to claim the entire amount, especially if that means one of you would then claim a bunch of itemized deductions while the other one picks the standard deduction. Be that as it may, that person needs to actually pay the whole amount. Now remember that money is fungible. If the two of you were to come to an agreement that Partner A will pay the mortgage while Partner B will pay for groceries, gas, utilities, etc., such that you each pay about the same amount overall, I see no reason that wouldn't be legitimate.