Jews in Italy There was a large Jewish close-knit community with a synagogue in Rome, as well as other major capitals in nations around the Mid-east, ever since the Diaspora, when Jews were expelled from Judea in 586 B.C. The fact is substantiated in the New Testament by the incident of the expulsion of all Jews from Rome by the Roman emperor, Claudius. Ironically, they were expelled because of rabid arguments over one called Chrestus (Christos, probably; Acts 18:1-3). No doubt these heated discussions were over the requirements of the Law, or the cessation of them due to the ministry of Christ.
But the Jews were quite mobile, transitory, because most of their occupations dealt with trade, commerce, banking, etc. Often they weren't allowed to be large land-owners, or permanent residents. Occupations that were available were often those not requiring residential citizenship. So even though one emperor would persecute them, they often found relief with the next one...and returned.
Under these circumstances, when Paul wrote the Romans, there were no doubt Jewish believers, rabbis, and Christians who were ex-Jewish synagogue members...as well as ethnic Christians in Rome. There would have been no problem for church members to be conversant about the Jewish and Gentile aspects of Christianity...the Gospel...the End of the Mosaic Law...and the amazing grace of God. The Roman Empire was quite efficient concerning economic, religious, and military intercourse. The Romans would have been informed about the schisms in Jerusalem concerning grace and law.
Conversant That the Roman Christians (Jew and Gentile) were conversant with with Jewish and Christian dogma is evident by a look at Paul's vernacular in the introduction to the Roman letter. Paul assumes the recipients of the letter know about: scripture, prophets, Gospel, Son, seed of David, Spirit, holiness, resurrection, Lordship of Christ, grace, apostleship, faith, saints, etc. (1:1-8) They would have known about the conflict between the Gospel of grace and the "Works of the Law" (legalistic ethos)...but they needed additional teaching, and encouragement lest they fall into the error of the Judaizer who had harassed the young churches of Christendom. Hence the book of Romans.
Legalistic Ethic According to research by J, Julius Scott, Jr. (Jewish Backgrounds of the New Testament), legalistic thought is found in intertestiment writings such as 2 Esdras, 2 Baruch, probably "Some Precepts of Torah (4QMMT). And a source of evidence of this Jewish thought must be recognized: the New Testament!
This legalistic ethos is found in dialogue of the Jewish scholars, as well as the common folk, with Jesus:
As Jesus started on His way, a man ran up to Him and fell on his knees before Him. "Good teacher," he asked, "what must I do to inherit eternal life?" (Mark 10:17)
Then they asked Him, "What must we do to do the works God requires?" Jesus answered, "The work of God is this: to believe in the One he has sent." (John 6:28-29)
Compare the story of the Prodigal Son, and the self-righteous elder brother (Luke 15:29); and compare the story of the self-righteous Pharisee praying in the Temple (Luke 18:p)
Precept of Torah As a note, the Hebrew,ma'ase ha-Torah, (Precept of Torah, 4QMMT) is the equivalent of the Greek, erga nomou (LXX). This is the wording which is the subject of our posted question, Romans 3:20. (And is also found in Galatians 2:16, 3:2,5,10.)
Would the Romans have had to look to the rest of the letter to explain what he meant? To answer this one must recognize that Paul had already broached this topic previously in this letter:
...they are a law unto themselves, since they show the that the work of the law are written on their hearts... (Romans 2:14-15,ergon tou nomou)
This shows that the Romans are familiar with matters of the Law, before they even read 3:20! (ergon nomou) But if they overlooked the mention of the "Work of the Law" back there, they would not have had to go very far to have an explanation: the very next sentence!
But now apart from the {works of} the Law, righteousness of God has been manifested... (3:21)
But if they even then did not get the message of Romans 3:20, Paul continued to deal with the subject over and again: 3:27, nomou, ergon; 4:2, ergon; 4:6, ergon; 9:32, ergon nomou; 11:6 ergon. Works! Works! Works!
The Christians in Rome--even the Gentiles--would have known of the "legalistic ethos" of the Jewish religion, and the issues it raised for new Christians. The Jews who surrounded them, and the Judaizers in their midst, would have kept this issue in the forefront of the Christian congregation.
Even today, "the call to grace, which requires knowledge that the human is helpless to contribute to salvation, goes against the grain of the "can do" outlook of many, particularly in the Western world. It is a humbling experience to have to accept grace." (op. cit. Julius Scott, Jr. p. 277 ftnt.)