Should I only claim half of my outstanding mortgage principle if I co-own my house with my spouse and we are filling married separately?

Yes, if you both are itemizing and each pay half the mortgage, you would each claim half. 

When filing separately, you must both Itemize or both take the Standard Deduction. 

You can agree between the two of you how much each can claim.  It can be 50%, 100%, or anything that the two of you agree to and that adds to 100% together.

As indicated by @KrisD15, you can deduct the mortgage interest in any way you both agree.  If you do not agree, and you did not equally make the payments, you should use only what you actually paid.  Be sure to have your proof of payments should you need them later.

The IRS would allow a deduction for the person who made the payment if you do not agree between you.

• Publication 504

@attheboulevard