Shooting films silently and adding the sound afterwards

The wikipedia page for "La Strada" states (emphasis mine)

As was the common practice for Italian films at the time, shooting was done without sound; dialogue was added later along with music and sound effects

So it was both

1. very common practice
2. specially italian - many western movies were also done that way.

It still happens, depending on other factors (noisy film location as per this question, or as it happened with Mad Max: to replace the Australian accent by an American one). Not necessarily filmed silently, but the audio is replaced afterwards.

I'm going to stick with the Italian flavour of filming without a soundtrack since director Federico Fellini used it and information is more readily available on one celebrated international director than a dozen others with less fame.

Filming without a recorded soundtrack was common among Italian filmmakers. Although I have no sourced proof, I strongly suspect that this was done to increase the pool of available on-screen talent since filming without sound did not require any one actor to speak a particular language. Additionally, multiple voice-over soundtracks could be cheaply produced in a variety of languages for international release.

However, with regard to the referenced example La Strada, director Federico Fellini did not have the actors speak their lines in his/her native language. Rather, a system known as 'number system' or 'numerological diction' was employed.

The actors would recite a series of numbers; e.g. 1, 2, 3, 4, etc. For 15 spoken words, they might count up to 30. This was used so Fellini as director could get them to redo a scene with alternative physical actions. Example, "Do it as you did it before but this time at 25, turn and bow your head to the left."

When employing this film-making protocol, directors such as Fellini could keep up a running commentary; shouting direction to the actors on-the-fly since there was no reason for a 'quiet on the set' environment.

The post-production voice-over synching was noticeably lackadaisical in Italian films.

Composer Michel Chion has observed that Fellini particularly exploited the tendency of Italian films of the post-war period to allow considerable freedom in the synching of voices to lip movements, especially in contrast to Hollywood's perceived "obsessive fixation" with the matching of voices to mouths: "In Fellinian extremes, when all those post-synched voices float around bodies, we reach a point where voices--even if we continue to attribute them to the bodies they're assigned--begin to acquire a sort of autonomy, in a baroque and decentered fashion."[44] In the Italian version of La Strada, there are even instances when a character is heard to speak while the actor's mouth is shut tight.
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I'm not sure if the lack of synchronization reached the level of a B grade Kung-Fu or Godzilla movie from the 1960's but it certainly seems that synchronized voice-overs were not a high priority.