They will certainly need the notice of their tax number, ΑΦΜ, since that appears on the bill and I think they might need a copy of the official rental agreement. I am not certain because, when I rented an apartment in northern Greece for almost 10 years, the landlord kept the electricity bill in his own name and simply gave me the bills to pay every two months. Similarly for the water. The telephone was in my own name. To me "taking responsibility for payment of the electricity" simply means agreeing to pay it not necessarily having it in your own name?