It's not entirely clear. First of all, this question is confounded by literacy but being a literacy/illiteracy binary choice, but a continuum. Especially in the past, people could often read but not write. Still today, people might be able to read a menu or a sign, but have trouble reading a novel or a transcript of a speech. There's also another difficulty with liturgical languages, like Hebrew, where it's common for people to be able to read the words printed on the page but either not understand them at all, or only get the gist of the content. This would've been especially common where people's native language was written in Hebrew characters, which was the case in many Jewish communities until recently.

Anyway, in the late ancient period it's fairly clear that a significant proportion of the population was illiterate. Being able to read the Torah is a skill, for example. At the same time, having written documents floating about is sort of assumed (with written marriage contracts and the like), as is literacy among the Rabbis, who were a scholarly elite but not necessarily an economic elite. Comprehension of Hebrew was also limited, since there's discussion of translating parts of the liturgy to vernacular languages (again here Rabbis seem to have understood Hebrew, and been able to write in it, but they were not the norm).

But, you asked about the high/late Middle Ages. Most Jewish texts we have in general tend to be scholarly religious texts, which are not about daily life, but do offer insights into it. One interesting passage relating to literacy is from Kuzari, a theological work. It is framed as a dialog between a Khazar considering conversion to Judaism and the author, R' Yehudah HeLevi. The Khazar asks the reason why Jews sway when reciting Hebrew texts (a practice today usually called shuckling, from Yiddish). The answer given is that there was a practice wherein the liturgy would be written in one very large book, which would be placed on the ground, and several people would read from it at once. They'd bend over, read a few words, bend back up, etc, which over time developed into a habit of swaying.

The important bit from this is that it assumes that most synagogue attendees in R' Yehudah HaLevi's time and place (11th-12th century Spain) were able to read a Hebrew text. It's not clear whether they could understand what was being read, but they could read the letters and say them. If they were able to do that, presumably they would be able to use that skill to read with comprehension in another language using Hebrew letters, such as Judeo-Spanish or Judeo-Arabic..

There are clues in other texts, too. Some medieval texts discuss whether a Torah scroll is valid or not, and in that context children are used as testers, so-to-speak--if a child can read such-and-such, it's valid. In this context it seems that children are assumed to have knowledge of the letters and their shapes, but not necessarily be able to identify individual letters by the word they're found in--an adult could identify a word, and therefore correctly identify a misformed letter, but a child couldn't.

But, the best selection of sources we have for everything Jewish for daily life is from the Cairo Genizah, a huge document cache from Medieval and early modern Cairo. It contains a whole lot of fascinating things, one of which is information on teaching children to read and write Hebrew (and, to a lesser extent, Arabic). It seems that children were sometimes educated with whatever written material could be scrounged up, with groups sharing manuscripts, but there were also booklets for practicing forming letters, a technique which children in modern times also used (including myself). There is no emphasis on learning to read or write words or phrases, which suggests that average people may've only partially understood Hebrew, or if they could read it, perhaps could not write down thoughts in it even if they could form the physical letters. And it is suggested too by these texts that it wasn't just wealthy children learning, but regular Jews learning to read (though it does seem to have only been boys. There is occasional evidence of women's literacy, such as what's possibly a woman's prayer book from Italy, but it seems to have been rare). There seem to have been groups of children learning, not individual tutoring (which also happened), and children from families who owned no books learning to read. Of course, this is a fairly urban cosmopolitan community--perhaps literacy was less widespread more peripherally.

So it does seem that the population as a whole was literate. That doesn't mean everybody was, and it doesn't mean everybody could understand the Hebrew they were reading--people may've had more understanding of their local language written in Hebrew letters. The fact that that was a common practice is evidence of literacy in Hebrew characters--Jews who spoke Spanish were writing down Spanish in Hebrew characters they knew and could use, rather than Hebrew script being the sole purview of those reading and writing in Hebrew for religious purposes. But, ability to read and write seems to not have been restricted to the elite.