Gregory of Nyssa (335 – c. 395) has a passage describing these discussions in Constantinople.
The whole city is full of it, the squares, the market places, the cross-roads, the alleyways; old-clothes men, money changers, food sellers: they are all busy arguing. If you ask someone to give you change, he philosophizes about the Begotten and the Unbegotten; if you inquire about the price of a loaf of bread, you are told by way of reply that the Father is greater and the Son inferior; if you ask “Is my bath ready?” the attendant answers that the Son was made out of nothing
To what extent can this be taken to be the case across the Empire? Are there any other reports of this kind of popular discussion? How did it vary? Why were these seemingly obscure discussions important to common people?
Ah yes, Gregory's famous passage in 'On the deity of the Son and Holy Spirit, at on Abraham'. It's one of my favourites!
No this is not likely an accurate description - this passage is famous precisely because of it's humourous exaggeration. It reflects a very particular, and important, discussion of the relationship between Father and Son in the period 350-390, which was a major dust-up in the church and had basically engulfed the whole empire. Constantinople, obviously, was a central place for this kind of discussion, and it's not unwarranted to think that the everyday person had an awareness, and possibly an opinion, on the issue.
Why were these seemingly obscure discussions important to common people?
Perhaps it is simply perspective, but these discussions only appear obscure to us. They were vivid, important discussions taking place at the time, which people understood to be fundamental questions which you could not get wrong without being a heretic, throwing the whole church into heresy, and going to hell. Most people have difficulty understanding that this is more serious than life and death for believers - being wrong about whether Jesus is truly God, is the difference between worshipping the true God and being a blasphemous heretic responsible for destroying the possibility of salvation itself. So these are not 'light' matters, and hence they were not solely the preserve of ivory-tower academics.
The great difficulty of your question, though, is working out how extensive awareness and debate about such issues was among ordinary people. It's an historical question exacerbated by the fact that there is virtually no documentary evidence written by 'ordinary people' that speaks to this question.