Rent a flat in Trastevere or Piazza Navona for a month?

Trastevere is a large neighborhood. It will help to know where exactly the apartment is located. The area near Santa Maria in Trastevere is nice and full of life, but has tourists. As you move away, toward the south of that area, there are fewer tourists and more local Romans, but you are moving away from the historical center. The no. 8 light rail tramway runs along Viale di Trastevere, if you are close to that corridor, you can get to Piazza Venezia quickly..
Piazza Navona is in the middle of the action for sure. But that is Tourist Central, so Italian will not be the prevalent language you hear, and in September, it will be noisy till late at night with tons of visitors walking around or eating at restaurants and cafes, unless the apartment is hidden away in an alley away from the main drags.
A quieter residential neighborhood nearby is Prati. Not far from the center and better served by the metro.

I agree with Roberto. Though he surely has far more experience than I do, I'd pick Prati for an extended stay. Better markets and shops, less touristy restaurants, quieter streets and an easy walk across the river to the centre. I stayed there for a week last year, walked back and forth every day with no problems, but also tried the bus a few times which was very easy to use. Once I figured out how to insert the bus tickets into the machine for validation, that is. A number of bus routes stopped half a block away. I was very near Piazza Cavour and Castel Sant'Angelo. I recommend you look in that area.

Trastevere has been "hip" for some time now. "Hip" almost always means "young, loud, brash, and noisy" unless the word is followed by "replacement." Many college students hang out on the street there, drinking and screaming until all hours. I enjoy my visits to Trastevere, but I doubt I would want to live or reside there overnight. I've never viewed it as central.

The Navona/Pantheon area is popular because it is, in fact, central, especially if you plan to visit the top ten most popular sites. Most people like to discover Rome on foot, so being central does cut down on wear-and-tear of the feet. There are plenty of locals still living in this area. I've never had difficulty finding someone to speak with in Italian. But it's true, both places are extremely popular with tourists these days.

If you want quiet and don't want to be surrounded by tourists, you need to look at neighborhoods that are less central and definitely not "hip."

As someone who lives in Rome, I cannot but agree with the previous comments. The bottom line being, if you want an authentic, quiet-ish (heaven knows how busy and noisy can this city be!) experience in the city, you would need to watch further away from the city center and into districts that are not as swamped by college kids or travellers. Another advantage would be prices - as in, they'll be way lower (which can be an interesting factor since you mention your trip is weeks away).

While not completely devoid of tourists, Testaccio (opposite the southern tip of Trastevere) could be an interesting option, as is Monteverde Vecchio (the hill above Trastevere). Both are well connected to the city center, the latter quieter than the former. Or Flaminio, north of piazza di Spagna. Just throwing these out there!

Another vote for Prati. While we had a hotel versus an apartment, we enjoyed that area for its more residential feel while still being not far from center. We're hearty walkers so went everywhere on foot but as Roberto mentioned, some of that area is well served by Metro: there was a station (Lepanto) just a block or two away.

Friends rented an apartment near Campo d'Fiori last February. They liked the residential area as it was not touristy and there were all the shops they needed for food...as well as a cafe in the small piazza. unfortunately they are off cruising so cannot get more info but might be an area to consider.

Thanks everyone for the feedback! Alas, not long after I posted this, the owner rented the Trastevere apartment to someone else. I did look in Prati, Monti, above the Spanish Steps, and a little in Testaccio and Aurelio, but couldn't find much for one reason or another (availability, budget, suitability, location, etc.). I did not decide to go on this trip until very late, so the inventory was very low as I have found out that Sept/Oct is also a peak time. Interestingly, I had some difficulty finding people who would rent for a month--it was often more short-term, maybe a week at most, or longer term, 3-6 mo minimum. I figure I can do some reconnaissance work for next time where I will plan much more in advance. Thanks again!