Most famous neighborhoods are touristy in some places and authentic in others, often within a few hundred meters of each other. The skill is recognizing the line in real time and crossing it.
Trastevere is a perfect example. The two main piazzas — Piazza Santa Maria in Trastevere and Piazza Trilussa — are tourist territory: overpriced, mediocre food, hassling waiters out front. Walk three streets in any direction and you are in the real Trastevere: family-run trattorias, locals walking their dogs, cobblestones with no English menus.
The same dynamic operates in Le Marais (avoid Rue des Rosiers around lunchtime, walk one street north for the real neighborhood), in Notting Hill (Portobello Road on Saturdays is a tourist crush, the surrounding crescents are perfect), in Greenwich Village (Bleecker Street is now a chain-store strip, the side streets are still magic), in Plaka in Athens (the main pedestrian thoroughfare is a souvenir shop, the streets one level above on the Anafiotika hill are extraordinary).
Our heuristics: (1) any street with multiple souvenir shops in a row is a tourist street, leave it. (2) any restaurant with a host outside trying to lure you in is a tourist trap, walk past. (3) any street where every menu has photos and English-only is a tourist street. (4) a quiet residential street with one café full of locals is almost always better than a busy commercial street with ten tourist restaurants.
The good news: in almost every famous neighborhood, the authentic version is a five-minute walk from the touristy version. You just have to be willing to walk.