If I had to guess, I think that it would be pretty safe to say that the mode was zero. That means that a majority of ancient Israelites could not read and did not personally own any books. But some percentage of ancient Israelites could read. Some percentage of them did own literary texts or works of knowledge. Again, the absolute percentage need not be large, but chances are that if you were privileged enough to read, you probably wanted to possess something to read.
Unfortunately, we cannot answer that question, but we can get some idea by looking at ownership of literary works in the Neo-Assyrian empire. SAA VII 49-51 are three lists of tablets owned by various individuals in the Neo-Assyrian empire. The texts are somewhat fragmentary, but they typically list the works and how many tablets in the work, and a summary of the number of tablets accompanied by the name of the individual. Taking the entries where the total number of tablets owned is more or less intact in all of the texts, we get the following list (in ascending order by tablet):
- Aplaya owned 1 tablet
- Mushezib-Nabu owned 1 tablet
- Tabni owned 2 tablets
- Nabu-balassu-iqbi owned [x]+2 tablets
- Nabu-shum-[. . .] owned 5 tablets
- Assur-mukin-pale'a owned 8 tablets
- Shamash-eriba owned 28 tablets
- Nabu-shakin-shulmi owned [x]+37 tablets
- [...] owned 100+[x] tablets
- Arraba owned 185 tablets
- Nabu-nadin-apli owned 188 tablets
- Nabu-[. . .] owned 435 tablets
I would expect ancient Israelite personal libraries to show a similar spread. Some would only own a work or two. Some would have several. What is somewhat surprising is that multiple individuals had extensive libraries, the equivalent of dozens of scrolls. We should suppose that ancient Israel would be the same.
It would be nicer to have a larger sample size. It would be nice if we had equivalent lists from Israel. But based on the information we do have, highly literate individuals with large libraries are known from pre-exilic Israelite times.