In Indiana, you have two separate documents. One gives you ownership of the property and the other says that you promise to repay the money you borrowed to purchase it to the bank and they can hold the house as collateral. The first is the deed, and the second is the promissary note or the mortgage.
If you both are applying for the mortgage together (and both qualify for it), you probably want to be on the deed. Why would you pledge to pay back money on something you have no ownership interest in?
If only one of you qualifies for the loan, you can both have an ownership interest. There, one of the two owns a house, but is not financially liable for it.
The exception to this is generally seen after a divorce. Both the parties are on the mortgage (and financially responsible), but only one has ownership in it per the divorce decree. If the party with ownership doesn't make the payments or the home goes into foreclosure, both parties can suffer the consequences.
I hope this answers your question. I'm not an attorney, but can refer you to one as needed. If there is something I can do to assist you further, you can visit my website and contact me from there.
PS. I am assuming you mean mortgagor, as the mortgagee is the bank or lender holding the mortgage. If I am wrong, it would obviously change the answer somewhat.