What do I put down for the cost of my rental asset if I co-owned it with my sister? 50% of the debt we took on, or the full price?

Neither.

assuming your split is 50/50, your cost is half of what was paid for the property in total as identified in the sales contract.

While you each can report your share of ownership, income and expenses, it would probably make more sense to report all rental income/expenses/assets on on 1065 partnership return. It can provide more flexibility in the future should things change that would call for it. (such as one of you being unable to pay your share of expenses for any reason up to and including death.)