I don't think the question of "are these two words, or one word with two forms" is particularly interesting linguistically, at least, not if you're basing the answer on the intuitions of illiterate people. If they're illiterate, they might not even have a well-defined concept of "what is a word" (even linguists don't agree entirely). To give an example from English, it seems like asking if illiterate people consider a and an the same word. I may be misunderstanding what you mean here. In any case, I think the interesting part is how people actually use forms.
There may not be a single answer here, since "liaison" in French covers various phenomena that don't all behave in exactly the same way.
My understanding is that the liaison pronunciation of "c'est" as /sɛt/ before a vowel-initial word, even though traditionally described as "obligatory," is often not truly obligatory even in adult speech. View the following page: http://www.projet-pfc.net/le-francais-explique/la-liaison/types-de-liaisons.html I don't know of a detailed account of its acquisition and use. A somewhat contrasting case is liaisons between determiners and vowel-initial nouns, which are acquired relatively early and are robustly obligatory in adult speech (aside from production errors), so I will talk about them a bit.
Liaisons between determiners and vowel-initial nouns
Here is a paper that describes some data and theoretical models of their acquisition by French-speaking children: Liaison acquisition, word segmentation and construction
in French: A usage based account, by Jean-Pierre Chevrot, Celine Dugua, and Michel Fayol.
This paper supports a viewpoint that I have also seen in other literature on this subject (such as this paper by Bybee, whom the authors cite): rather than treating the liaison consonant as simply a word-final consonant, it argues for a "construction"-based model.
Here's how this works: a phrase like des filles /defij/ is considered to have a mental representation of "determiner + X" (/de/ + /fij/).
But, a phrase like "des amis" /dezami/ is not just "determiner + X" with a different word as the determiner (/dez/ + /ami/). Instead, the mental representation is postulated to be "determiner + zX" (/de/ + /zami/).
The paper terms this the "exemplar-based" model, because the hypothesis is that children first acquire the concrete forms of a noun that are prefixed by a liaison consonant (the lexical representation of a noun like ours would include /urs/ /nurs/ /zurs/ /turs/) and then learn the appropriate contexts where these are used.
The model that it argues against is the "autosegmental model" where liaisons are modeled as floating consonants attached to the preceding word (i.e. children learn underlying forms like /de.z/ for des and then combine them with vowel-initial lexical representations such as /urs/).
I haven't seen anyone advance a model where forms like /de/ and /dez/ are represented as two separate words, even for illiterate children.
Liaisons between masculine singular adjectives and vowel-initial nouns
In Concreteness in Generative Phonology: Evidence from French, by Bernard Tranel 1981 (a book with some nice discussion of liaison and other elements of French phonology) it is proposed that at least some linking forms of masculine adjectives used before vowels are suppletive. The specific examples given are bon /bɔn/ (normally bon /bɔ̃/) and bel /bɛl/ (normally beau /bo/). So in this case, they would be stored in the lexicon as separate forms. But this is not the same thing as being separate words, any more than the masculine and feminine forms of adjectives are separate words.
Native French speakers (even adults) sometimes make errors where they insert a liaison consonant that is not etymologically warranted. (Omitting to make an optional liaison is not considered a mistake.) I don't see how these could be due to literacy, since the consonant is not actually present in the standard written language. In addition, these erroneous liaisons are rare, so it's unlikely the speaker learned these particular erroneous forms by hearing and repeating them. The existence of these errors provides evidence that native French speakers do not just learn liaison consonants word-by-word; they learn general rules that they can extend to create innovative, non-standard forms.
The following examples are taken from Tranel, p. 228:
- ce règlement devra être révisé
'this regulation will have to be revised'
- en dépit d'une circulation que chacun continue à trouver fluide
traffic that everybody continues to find fluid'
- le nouveau commence à remonter
'the new one is beginning to come back'