You are each allowed to deduct the portion of mortgage interest and property taxes that you actually pay. Because you don't get any benefit unless your itemized deductions are more than the standard deduction ($14,600 for single and $21,900 for head of household), it is possible that if you split the payments, then neither of you will have enough itemized deductions to benefit (of course, this depends on your other deductions, including state income tax, gifts to charity, and the size of the mortgage).
You can do anything you want as long as you can prove it with reasonably accurate records. For example, you could draw up a budget showing that person A pays 100% of the mortgage and person B pays 100% of the utilities, food, and other household expenses so it balances out. As long as you stick to that budget, so your bank records match, the IRS should not have a reason to argue. If you want to split the interest and taxes, you could draw up a similar budget showing that and use a joint account, or you could open a third bank account just for the mortgage payments and fund it by transfers from your main individual accounts, or you could alternate payments. (Although that seems like a hassle or a risk of missing a payment, if you have to change the auto-debit info each month.) The main thing is to have a reasonable story, stick to it, and make sure the records reasonably match.
