Of course, it might have been either. But Luke 7:5 sounds as if the [at-least-partly-Centurion-financed] Synagogue building at Capernaum was that city's first-ever corporately owned, specifically allocated, communal construction. And since Capernaum was probably a hair better off than Nazareth, economically, it seems very possible that Jesus' family and neighbors actually gathered in a converted home or something similar.
(Seriously, my house church friends should NOT be getting excited about this - practical expedience was extremely common in antiquity. On the other hand, I do wonder if this being a common practice might help explain the relative dearth of early Synagogue finds in Palestine. But as far as my purpose here, I'm merely trying to be thorough.)
My only point is that we don't need Archaeology, in this case, to expect that the Jewish community of Nazareth held gatherings in some place or another during Jesus' early life. Whether they utilized part of a shared building or had somehow afforded themselves a proper meeting house, the Synagogue was the Community.
This well established fact was especially true before 70 AD, and clears the way for us to ask the next logical question. Given that there almost certainly was a Synagogue association/gathering of some sort in Nazareth, during the Lord's "silent years", to what extent did Jesus participate in it, and for that matter, what precisely did they do?
To be continued...
Series Update: The Nazareth Synagogue