In essence can they take possession of the other party's interest in the property by foreclosure if they did not sign the mortgage? If the person liable for the mortgage default? In this scenario
When property was purchase 2 people were on the deed and only one was responsible for the mortgage, mortgage was refinanced with the same bank under the same scenario 2 people on deed only one signed the mortgage documents
Look carefully at the mortgage documents. When you have A&B on title, and only A is on the mortgage, B is required to sign a Deed of Trust. This is different from signing a Note (which is the actual debt). The Deed of Trust is an agreement by all who sign, that the Lender is allowed to foreclose on the property in the event of a default.
However, I have seen many cases where the Deed of Trust has not been executed properly - which would be your only practical lifeline in a case like this. The Deed of Trust is the only thing that allows foreclosure (because it uses the house as "collateral" in the event of default), so if this is not executed properly, the house is not collateralized properly, and no foreclosure rights exist.
This is a very, very common situation. Everyone who is on the deed, as an owner, should have signed the lien instrument. If not, the lender may be only partially secured, or totally unsecured. For example, if a house is owned by a married couple, but only the husband signed the deed of trust, the the lender is not secured. And if two unmarried folks are on the deed, and only one signed the deed of trust, the bank may be partially secured, and could foreclose only on on person's interest. It will only take one or two hours of a lawyer's time to figure all this out for you.