No the arbitrator only steps in if something goes wrong. In normal case the traders do the transfer and confirms send and receipt of the fiat.
Fiat si always transferred out of system, like in LocalBitcoin. You go to your banking webpage and do the tx. The other will check on his bank web page if he received it. The app only exchanges the bank details data needed to do the tx and to verify the receit.
You only can trade BTC vs. Fiat or BTC vs. Altcoin. Not Altcoin-Fiat or Altcoin-Altcoin.
There are some more videos at the videos section on the webpage.
Wouldnt this open an issue on trust while the transaction on the bank are going through, in other words, if I send the tx, what guarantees me that the recepient wont just log out and leave? Or say that he doesnt see the transfer or argues that there was an extra fee and he wont have the full ammount. I see a lot of space for disruption on this side and no recourse in case something doesnt go right.