If you share a debt with someone else this is called ‘joint and several liability’. If you take out a trust deed the other person can be pursued for the whole debt. The joint debt is listed in the trust deed but no payments are made towards it by the trustee. It is possible for you both to have a trust deed but you should both discuss with the trustee whether or not this would be advantageous.
If the relationship has broken down then joint debts are one of the issues that should be discussed as part of all the financial circumstances of the relationship.