I can talk about Greece which I am more qualified. In Greece we have two different kind of remains:

1. One are physical remains. Among them chiefly we have the synagogue of the island of Delos, a major trading hub of that era. Dated around 150 BCE there are controversies about the interpretation of inscriptions dedicated to ΘΕΟ ΥΨΙΣΤΟ but the consensus is that it was either a proper synagogue or a private dwelling used for a congregation. Samaritan inscriptions lend credence to this interpretation although shy from being the smoking gun evidence.

There are also other synagogues in greek islands like the 3rd century mosaics belonging to the floor of the Aegina synagogue but I do not know whether chronologically this would fit to your requested timeframe.

Other physical testimonies speak about synagogues which didn't survive time. Eg a sarcophagus from 3rd century CE in Thessaloniki asks that the violators of the grave pay an astronomical amount to the synagogues of the city pointing out to multiple locations that didn't survive.

One of these synagogues could easily be a 3rd century Samaritan synagogue, whose dedicatory inscription was found in the centre of the later byzantine jewish quarter of Evraida.

In other cases we have buildings which weren't built as synagogues but were used as such at one point. The Μητρώο inside the Agora of Athens was a public record building but at some point from the 3rs century onwards was used as a synagogue with some decorations still surving. The building itself was destroyed later during barbar invasions but menorah and lilac decorations survived.

2. The main literary sources is the Letter to the Thessalonians where Apostle Paul speaks about his experiences among the well organized and powerful Jewish communities of Thessaloniki and Veria.

The Etz aHaim synagogue/kehila claimed that it was founded then and indeed followed the Romaniote minhag of the Byzantine Jews. Incidentally topographical evidence of the position of the synagogue in the 19th century would imply that the position was ancient enough for the walls of the city to have molded around it than the other way around, lending credence to its ancient origin.

The same way about Corinth were Paul preached at the synagogue. While the building has not survived, a 5th century inscription speaking about "synagogue of the Jews" and menorah decorations have been found.

So in Greece we have remains of synagogues which still survive like in Delos or Aegina. In the Ancient Agora of Athens we have a building of mixed use. In other cases only inscriptions survive talking about multiple multi-denomination synagogues in Thessaloniki or the one in Corinth. And indirectly certainly in places like Veria Jews must have certainly have had their own congegations.