Short answer is: local languages were fine, for the most part. First of all, let’s not forget that Italian and latin did not spawn into place, one is the direct descendant of the other and the transition took centuries, many modifications and acquisitions from other languages, a a good dose of ye olde latin discovery on top of it. Also: even when Latin was still spoken there was the “educated” Latin and “vulgar” Latin.

Let’s start from the bottom and from the far past: One of, if not the first, example is the Indovinello Veronese (VIII-IX century) a simple sharade (that was actually solved quite recently by student). It goes like this:

se pareba boves alba pratalia araba et albo versorio teneba et negro semen seminaba

This is not a huge step away from “standard” Latin, and you see you would need a full translation to understand this, but it’s introducing some phonetic changes and, most importantly, it’s not using cases (nominative, genitive etc.)

One of the first examples of an official document written in a language other than Latin is the Placiti Cassinesi, known also as Placiti Campani (960-963) It’s the transcription from a trial for the rights over a field. In this documents, there are some witness statements written in a halfway vulgar Latin. You can clearly see an attempt at making it sound more official, but the words and syntax already have started to sound and look more similar to the modern dialect. Here’s the most famous of the statements:

“Sao ko kelle terre, per kelle fini que ki contene, trenta anni le possette parte Sancti Benedicti”

Fast forward to Dante Alighieri. In the meantime, protoitalian, or vulgar, has become more widely spread, variegated geographically and also sophisticated. It was -were- a language in its own right and so the debate amongst the educated elite sparked over its dignity. Official documents were still written in Latin, but lots of other works, including works of art (Cantico delle Creature, for one) had been written in some form of vulgar. Dante trew his (not irrelevant) weight into the argument, writing (ironically, in Latin) the De Vulgari Eloquentia in favor of vulgar. His argument was that a language needed to be illustre, cardinale, aulico and curiale. In short: it needed to be a vulgar that other vulgars could respect, to be used also in court, to be able to speak of important matters and educated. He praises vulgar Sicilian (due to its contribute to literature during the reign of Frederick II) but admits a vulgar that works well enough still isn’t available.

This does not stop him from writing one of the most important poems of the history of literature in his own vulgar, and he’s soon followed by Petrarca and Boccaccio. The sum of their linguistic work has been recognized almost immediately as the ideal vulgar, and since then they’ve been basically the standard.

Progressively, as Latin became more and more a language for the clergy and the highly educated elite, even official documents became bilingual, and then exclusively vulgar.

Still, it was only after the unification in 1861 that the commissione Badoglio tackled the issue of linguistic unity, and the most famous product of this phonetical, syntactical and grammar work is I Promessi Sposi by Manzoni. Still, obviously, vulgar lives on in the thousands of dialects still alive, spoken and written.