I will answer only about Italian language, since I don't have direct access to French sources.

"How would all these regions communicate with each other?"

Well, the illiterates simply didn't , while instead all the educated people knew Italian and used it, although a lot of them not really well. The people who knew Italian also often didn't find it natural. For example, Manzoni (1785-1873), one of the greatest Italian novelists (and big "influencer" of the Italian language) always said that for him was easier speaking French than Italian. To explain better the situation, I will translate for you a portion of "Epochs of the Italian Language" by Ugo Foscolo. Circa 1825 (circa 45 years before Italian unification).

We know from experience that neither an illiterate Neapolitan can understand a [illiterate] Milanese, nor a [illiterate] Turinese can understand a Bolognese; nor four educated men from those provinces would be able to converse without misunderstandings if not using among them a sort of hybrid Italian that, even if participating in the regional dialect of the speaker, assumes anyway the desinences and the grammar of the literary language of the nation; and this national language even if it is not spoken, neither bad nor well, by the common people is nevertheless understood, more or less, even by the lowest plebs.

Now, you may ask, how many people spoke Italian at the time of the unification (1861). Well, Since we don't have a linguistic census we must rely on reconstructions, and we have 2 wildly different reconstructions based on different parameters:
a pessimistic reconstruction and a more (relative) optimistic one.
The pessimistic one was done by one of the most famous Italianistic, Tullio De Mauro. He considered as Italian speaker all people from the Tuscan region, all people from Rome and zones nearly it (The roman dialect, in a sort of misterious way, got "Tuscanized" in the 16th century ) and all people that had a post-elementary education. Doing so he calculated that about 2,5% of the Italian population was able to speak Italian.

Arrigo Castellani opposed this reconstruction, arguing that should be included also the clergy, those who had a domestic education and most importantly also people from the other part of Latium (thus not only Rome) and also people from the Umbria region and the Marche region, since according to him those dialects are similar enough to the national language and thus should be included. He thus gives a number of 2 million of Italian speakers, that were around the 10% of the population.

"How did the French and Italian governments make their respective languages so widely spoken?"

A great obstacle to the diffusion of the national language was the great rate of Illiteracy, at the time of the unification between the 75% and the 80% of the population was unable to read and write. (and in some regions almost the 90% of the population was illiterate). With the exclusion of the first ten years after the unification, in which the new state was in a dire situation, the government started to invest in education with quite good results, for example in 1911 only the 46,20% of the population claimed to be Illiterate. ( in 1861 78% of the population claimed illiteracy and in 1871 the 72,96%). In 1877 Italy, with the Coppino Law, made "inferior elementary schooling" (6-9 years) mandatory( Although it didn't enforce it that much).
The increase in literacy wasn't the only factor. The centralist view of the state, the new mandatory military conscription and the diffusion of newspaper and other media widely increased the number of Italian speakers.

Before the 1950s the rural zones remained quite excluded by this improvement, but a new object widely changed the situation: The Television. Not only indirectly but also in a more direct way. Between 1958-1966 aired the program "Telescuola" ( Tele-School) which aimed to give a medium education to the people who had not middle school in their towns or were not able to go to it for economic reasons and between 1960-1968 aired "Non è mai troppo tardi" ("It is never too late") a widely followed educational program targeted to illiterate adults and elders. (It is estimated that this program helped 1 million and half of people to obtain an elementary education licence).

"Was this pattern of language commonplace amongst european states?"

Well, yes and not. Especially in that period a lot of European nations faced problems relative to the national language, but those problems had their peculiar and unique connotations. Also, due to geopolitical reasons, the timeline of those problems differed among the different nations and countries.

"Why were these languages chosen instead of some other language in [...] Italian regions?"

The Italian language had a quite peculiar history. Since it is an extremely long and complex history I will be forced, due to time and space issue, to do an extreme oversimplification. In the late middle age, due to the work of 3 Tuscan men: Dante (Considered the"father" of the Italian language), Petrarch ( One of the greatest poets of all time, extremely important figure in all of European literature. His poetry was imitated literally for centuries in all of Europe.) and Boccaccio (Famous for his "Decamerone", he was the main influence for Chaucer's "Canterbury's tales" ) educated people from Italy decided that the language spoken in Tuscany should have been the basis for a common Italian language ( In fact, up until the 20th century, Italian was also commonly called Tuscan, and before 16th century it was rarely called Italian) but people disagreed on what should be characteristics of that language. Some argued that should be used only the language spoken in Tuscany by the educated people, some argued that should be used only the language from the city of Florence, some argued that should be used only the words most near to Latin, some argued that should be used only the words and the grammatical construct more near to the other regional languages, etc. etc.

Finally, in the first half of the 16th century, most people seemed to agree on the proposal of Pietro Bembo to base the poetical language on the one of Petrarch and the prose language on the one of Boccaccio. But, Gian Giorgio Trissino (Famous for trying to add more letters to Italian, such as ω , ε , ζ , ç , ʃ and a differentiation between i / j and u / v. He was successful only in the last 2 points.) objected the Bembo's proposal and argued that the national language should be a sort of mix of the various languages spoken in the various Italian courts and that it should be less archaic.

Educated people then chose mostly a compromise between the 2 positions (Favouring slightly Bembo's position). Of course, being Italian, they continued to argue on various linguistic questions for centuries and centuries but at least they did it using a language with the same foundations. With the advent of the 20th century the language underwent a sort of homogenization and the "Questione della lingua" (Problem of the language, as it was called) stopped definitely.