As a general rule the average lay person in early Christianity knew less about theology than the bishops did. However, in regards to the various Christological and Trinitarian controversies of the 4th and 5th century, the average Christian probably had at least some basic understanding of what was going on, and certainly cared, because these issues often had a personal impact on his/her religious life. There is not a source from the ancient world that gives us any kind of poll of religious feelings of average Christians, but a few points of evidence support my conclusion.

First, the Arian controversy (318-381) frequently led to the removal of bishops. These bishops were replaced with someone who supported a different theological idea. Early Christians seemed to have a close relationship with their bishops. They appointed them by acclimation. They cared for them as the Letters of Ignatius of Antioch seem to demonstrate. For a bishop to be removed this meant that the people's spiritual leader had been taken from them. Riots broke out in 339 and 356 in Alexandria when Athanasius was desposed on the two occasions.

The removal of a bishop was would also bring with it a change in preaching that an average Christian would have been aware of. In Ancyra, Marcellus was replaced by Basil, changing the theology of the city's leader from the idea that God was one person who morphed into another over time (modalism) to three separate, but unequal persons (homoiouisianism). Certainly the bishops presented their theology to the people and did their best to do so in a way that they would understand. For instance, Hilary of Poiters' Tractates on the Psalms, which are presumed to have begun as sermons, frequently refers to Christ as existing in the form of God and as the form of a slave. This is Hilary's technical language, based on Phil 2:6-7, to demonstrate the full divinity of Christ. It's clear to me that Hilary is instructing his congregation on proper Trinitarian theology. This certainly doesn't prove that his listeners understood all of the technical details, but it does show that this debate over the nature of God was actively being presented to the laity; it was not something they would have been ignorant of.

There does seem to be some indication that not only did this preaching increase awareness of the issues, but that many people did come to care about the issues directly. As u/Foojer points out, Gregory of Nyssa spoke to the interests of the people in the debates of the time:

“Everywhere, in the public squares, at crossroads, on the streets and lanes, people would stop you and discourse at random about the Trinity. If you asked something of a moneychanger, he would begin discussing the question of the Begotten and the Unbegotten. If you questioned a baker about the price of bread, he would answer that the Father is greater and the Son is subordinate to Him. If you went to take a bath, the Anomoean bath attendant would tell you that in his opinion the Son simply comes from nothing.” (Oration on the Deity of the Son and of the Holy Spirit)

(I hope I have not broken a rule here by incorporating his/her very good point here as part of my answer, if so my apologies and I'll edit this.)

Such concerns over theological issues continued into the 5th century. The Nestorian controversy was started over whether Theotokos or Christotokos was the appropriate title for Mary. The Christians in Constantinople had taken an affinity to calling Mary the Theotokos to which Nestorius objected. The ensuing controversy would have an impact on liturgy and prayers and as such would have an impact on the average lay person.

To be sure, the interest of the laity would not have extended to the highest levels of technical expertise. Getting to the specific example of OP, Chalcedonian Christianity versus miaphysitism, this is one of the most philosophically difficult issues of the early Church to illustrate. Regarding the Arian controversy, I am sure that the average Christian got the just of the debate, was Christ fully God or was he a subordinate deity? I have my doubts, even if the average Christian's cared, that they could articulate the issues at play when it comes to miaphysitism. Even experts today on early Christianity will shy away from articulating the technical issues at play because the distinctions being made by each side were rather small.