There are 2 distinct ways in which such an account can operate.
The 1st is a joint account - where you are both the legal owners of the funds & both legally responsible for any debts on the account.
The 2nd is a sole account - where the funds, either debit or credit, are owned legally by a single person. However, this sole person can authorise a mandate called a "Per Pro" that allows a 3rd party to make withdrawals, sign cheques etc., just as if the funds were there to do with, as they wished. However, on death of the sole account holder, the "Per Pro" mandate is automatically cancelled & that 3rd party no longer has any access to, or claim on the funds.