Here are two contradictory quotes from two pop history books I have read regarding whether Jews attempted to convert non-Jews in the Roman Empire:
Though Jews had had a strong presence across the Eastern Mediterranean for centuries, there is no evidence they attempted to convert Gentiles to their beliefs. Judaism was a national religon, not something to be pushed on outsiders.
Because Jews themselves were later on the sharp end of Christian and Muslim missions, there is a reluctance to accept that Judaism itself was a missionary religion. Yet as early as 139BC, Jews were being expelled from Rome for trying to convert Roman citizens. A little later, the great lawyer-politician Cicero complained about proselytizing Jews. Two emperors, Tiberius and Claudius, transported Jews from Rome for the crime of trying to convert Romans. Roman writers such as Horace, Seneca, Juvenal and Tacitus all discuss the issue. Later, the emperor Theodosius published ferocious decrees in the Christian era against anyone who attempted to make converts to Judaism.
So who is right here, and is there any consensus among historians?
In general evidence from this period from Jewish sources is relatively scant, or filtered through later Jewish sources that are difficult to untangle. And neither of those sources is particularly rigorous historically, so it shouldn't be very surprising that there are contradictions. Generally, though, there is some evidence of Jews proselytizing in the late antique period to an extent.
The whopping example is the interaction between Jews and Idumeans. The Idumeans were an adjacent group to ancient Judea. The Hasmoneans (the ruling dynasty in the late ancient period) conquered and forcibly converted the Idumeans, as recounted by Josephus (a 1st-century Jewish historian):
Hyrcanus took also Dora and Marissa, cities of Idumea, and subdued all the Idumeans; and permitted them to stay in that country, if they would circumcise their genitals, and make use of the laws of the Jews; and they were so desirous of living in the country of their forefathers, that they submitted to the use of circumcision, and of the rest of the Jewish ways of living; at which time therefore this befell them, that they were hereafter no other than Jews.
Most examples are much less stark. Josephus and other historians recount stories of Jews spreading their religion in less direct ways. Generally speaking, there's no point where they're "sending out missionaries"--there was no particular theological or political motivation to do so. However, there are a number of texts that describe regular Jews discussing their religion and, should things work out, converting others to it.
From later Jewish sources (i.e. Rabbinic texts) the early descriptions of conversion are a bit unclear, relative to the above. It is unclear how well their instructions were followed backwards in time. Generally, converts are to be dissuaded (in the sense of describing to them the drawbacks of conversion as "how do you talk to potential converts"), but there's nothing exactly stopping Jews from talking to non-Jews about religion and perhaps piquing their interest and eventually converting them, a description which doesn't necessarily clash with earlier descriptions of Jews converting people. During this time the Rabbinic establishment was not particularly powerful, so they wouldn't've been able to really set any particular conversion standards or practices anyway.
Later on, Islam and Christianity strictly forbade conversion to Judaism. This gained Judaism a general aversion to going out and telling people about Judaism. The fact that non-Jewish authorities in some areas, particularly in the Early Middle Ages, continuously banned conversion to Judaism suggests (at least to some) that Jews may've been converting people then, perhaps mostly servants and/or slaves in their households. But evidence of this from Jewish sources is mostly lacking, and it could simply be a Christian fear of other Christians converting to Judaism motivating this, not a real existence of conversion to Judaism.
So, it seems that in ancient times some Jews did encourage conversion. However, without a particular reason to do so, it does not seem this was part of an organized effort, just Jews telling non-Jews about Judaism and sometimes they'd decide to convert. Jewish suspicion of conversion is a somewhat later phenomenon, but even in ancient times, it doesn't seem Jews were itching to convert people either.