It would not be usual for the person you live with to stay with you whilst you are having contact. This can happen sometimes, but it is quite rare.
Whilst you are in contact the person who took you to the session may do a few different things. Often, they will not be in the session with you, but they might wait in the waiting room. Otherwise they might pop to the shops or find somewhere to buy coffee whilst they wait. The centre will always have a way to contact the person you live with, so if you need support it will never be too far away.