Who pays for what may depend on who owns the property and who can afford to pay for any outgoings. Ideally, you should discuss this and agree who will pay for what from the outset and include those decisions in a cohabitation or living together agreement. You are legally responsible for any bills, contracts, loans and credit cards taken out in your sole name, even if they were taken out for the benefit of the family. If your name is on the mortgage or the utility bills, the providers will expect you to pay the bills and not your partner.
If one of you already owns the property and the bills have been set up before the other moves in, it is still very important to discuss what each of you will pay and the implications this will have. This should be included in your cohabitation agreement.