It wasn't very clear what this process was for me. I read that you have to be in the AIRE system before you can get your passport. I received 2 emails from the LA consulate saying that my citizenship application had been processed. The first email stated that the would be sending the information to the comune for inclusion into AIRE. A few months went by and I decided to create my account in the FAST IT system. The form is a little personal information and you have to upload some documents to identify yourself. Once you're registered, you can look at your scheda consolare (consular file). There's a field that states Registration with AIRE. Mine reads "not registered".. still.
I also had moved since my application was submitted to the consulate and the time when I was going to apply for my passport. I was able to go into FAST IT and submit the form to update my address. I had to submit a copy of a utility bill to prove my address. I did receive a phone call from the LA consulate with someone who made sure that my address was changed and how we are still waiting for the comune to update my AIRE status, but the LA consulate was aware that I have been recognized and I can continue to get my Italian passport. Actually, a very pleasant phone call. Even ended the call with me saying "Mille grazie" and her saying "Prego, ciao!".