Staying away from modern sources, I will answer your inquiry by pointing to the evidence of ancient sources. For instance, most Palestinian Jewish communities had synagogues (even a little community like Nazareth where the common people like Jesus were invited to read from the Old Testament-- Luke 4:16-17). Each synagogue had a "Chazzan" (a teacher) who was charged with instructing young Jews in studying the Law. In the gospels Jesus at numerous times cites Old Testament scriptures that his listeners are familiar with -- even long passages. In fact, his critics even quoted scripture back to him at times and this prompted Jesus to say at various times, "Have you not read...." ( example, Matthew 19:4). Notice their response (Matthew 19:7). Another passage in at the end of the Gospel of John 7:41-53 is also an interesting commentary of the literacy of the common people. Pharisee leaders had to rebuke some of their own people for believing in some of Jesus' claims and works. Pharisee leaders had to appeal to the scriptures to support their own views (verses 42 and 52). In the latter verse, they even chided their Jewish audience by saying, "search [the scriptures] and look..." One more example is also telling-- In John 8: 1-11, Jesus saves an adulterous woman from a mob of zealots who want to stone her. They cite the Old Testament as a reason for doing this (verse 5). Jesus disarms them by writing an unknown message in the dirt that they read, understood, and then left (verses 8 and 9)...
But Jewish sources of the era also support widespread literacy. Josephus, a Pharisaic Jew, tells us in his autobiography, The Life of Josephus, 2, that he was "very proficient in learning," and began his own formal studies at age fourteen. In another Josephus work, Against Apion, 2, 26, he tells Gentile readers that Jewish parents were admonished to "bring those children up in learning, and to exercise them in the laws, and make them acquainted with the acts of their predecessors, in order to their imitation of them, and that they might be nourished up in the laws from their infancy, and might neither transgress them, nor have any pretence for their ignorance of them." In other words, if they were to keep the law, they had to know what the laws were-- only literacy can accomplish that.
Furthermore, the great Jewish sages of this era, such as Hillel, also admonished young Jews to study the law so that they could keep it. His maxim, "He who has acquired for himself the words of the Torah [Law] has acquired life in the coming world" (Mishnah, Aboth, 11,7), was directed at all Jews, not just community leaders.
Finally...widespread use of signage and inscriptions in antiquity. Do you remember that Jesus' cross inscription was in three languages for all to read (including Jews-- John 19:19-22). In addition, archeologists have uncovered many "signs" from this period of antiquity written in Aramaic-- the language of the Jews of Jesus' day. For instance, a few years ago...a sign in Jerusalem in Aramaic that read, "To the place of the trumpeting." This sign was once in the Temple complex and told where Temple trumpeters were to blow their horns. Other Aramaic signage has also been found in Jerusalem, Caesarea, and Herodium. The audience for these signs were Palestinian Jews. Also what are we to make of coinage, gravestones, and "bone boxes" also labeled in Aramaic? All of this implies a widespread literacy among the ancient Jews.