Q: Will the co-borrower, if responsible, have to pay off the entire loan, or just take over the payments?
A: Same thing.
Q: How can we tell whether or not the loan was written so that the co-borrower will have to take the loan over or not?
A: Ask the lender. Also, whoever signed the promissory note is responsible for the payments. From everything you have posted, the "co-borrower" (the better term is borrower) is responsible for the payment. Otherwise, the land will get sold out from under the owner(s).