They are jerking you around. The deposit is only dealt with by the owner that rented you the place:

"Arriving and Paying:

--It’s common practice for a Roman apartment to require a deposit to hold the property, and then ask for the balance in cash upon arrival, plus a deposit against potential damages. You should not pay your balance and deposit (against damages) until you have made a thorough inspection of the apartment and everything you will need for your stay. If you accept an apartment without doing this, you lose any leverage you might have, and in the eyes of the apartment rental owner and/or agency, have given your OK to its condition.

--make absolutely sure before your rep leaves you with the keys that you have towels, plates, and bedding for everyone, and have done a thorough inspection. Don’t be shy about this! Flush the toilets, try the hot water! Also make sure you have every emergency number they can provide.

Make sure you arrange in advance with the rep a time when they will come to do a final inspection and return your deposit. You should have their number, and they yours."