Does a co-borrower on a loan who isn't an owner of the property need to sign the mortgage as well?

No. A non-owner should NEVER sign a mortgage. A person who has no interest in the real estate should have no interest in promising to be responsible for the mortgage debt. If someone who does not own the property signs the mortgage they are promising to pay for property they do not own. If the primary borrower defaults, the bank will go after the co-signer.