First of all, we need to remember that access to Scripture, in terms of everyone having their own copy of the Bible, was just simply not a reality in anyway until well after Gutenberg invented the printing press in about 1450, when it became more economically feasible for the lower classes in society to actually own a copy of the Scriptures. And by that time we also have to take into account literacy rates. The phenomenon of every family (or individual) having their own copy of the Scripture is very recent in historical terms.
So "access" to Scriptures, from a historical standpoint, has to be defined differently.
Orthodox, from the very begining, have always translated the Scriptures and liturgy into the language of the people they are evangelizing. And as the Russian mission in the 1700s in what is now Alaska shows, they also worked very hard to make social and cultural connections to those peoples in communicating that Gospel.
Through history, the typical way that a Christian would have access to the Scriptures would be in the worship services. The Orthodox liturgy is itself almost entirely quotations of (or paraphrases of) Scripture, primarily Psalms. Additionally, large sections from the New Testament and the Gospels were read at the normal Sunday service, and during certain feastdays (such as the Saturday right before Easter) large portions of the Old Testament were also read. So for an Orthodox Christian, they were (and are) essentially immersed in Scripture and Scriptural allusions and phrases throughout the service.
Further, although literacy rates were higher in the East than in the West (generally speaking), even so, we have to remember these cultures were primarily oral cultures. They would hear and listen to the Scriptures and memorize them. The many examples we have from history of Christians who could recite large portions of the Scriptures is emblematic of that oral culture.
As to how individuals would "test" or "authenticate" specific biblical teaching, they would do so by comparing such teachings with what had always been taught (how ancient was the teaching, did it go back to the Apostles?), taught everywhere and by all (is this the universal teaching of Christians around the world?). There was no such thing as each person taking authority upon themselves to say what Scripture "really" meant--such is the definition of heresy.
Furthermore, Christians could tell whether or not a teaching was part of the apostolic deposit of faith by whether it "held together." Did a specific teaching make sense given what else a Christian knew to be true? Even in the ancient Christian world, everyone knew about "prooftexting" with Scriptures--pulling out a verse here and there to "support" an argument, without considering either the context of the verse or the context of the doctrines of the Christian faith. So, if Christians heard a particular teaching, and also heard someone quote a Scripture to "prove" that teaching, they knew that they could examine that teaching by comparing it to what they already knew.
The early Christians knew that no one has ever been given the authority to read the Scripture and on their own authority to declare that their particular teaching was right and everyone else was wrong. Further, the early Christians knew that there was no such thing as being able to understand the Scriptures on their own apart from entire Church's input and understanding. The early Christians knew that the Bible wasn't like the Koran (to borrow an anachronistic example) that God simply delivered miraculously to one man at one time. They knew that God gave the Bible through his Church (both Israel and then the Church). It was the Church that determined which books were Scripture. From the standpoint of the individual Christian apart from a canon there is no Scripture. So there is no reading of Scripture "on one's own." There is only reading Scripture within the Church, and can only be understood within the framework of what the Church has taught.
As to today: the Orthodox typically have continued interacting with Scripture as it was done for centuries. That means for individual Orthodox, and despite the vast prevalence of and ease of access to one's own copy of Scriptures, there is not a lot of personal reading of the Scriptures or of resources for the indivdiual Christian in reading the Scriptures. (To be fair, this is not uncommon among ALL Christian groups; biblical literacy isn't that great even among evangelicals.) What "saves" the Orthodox in this regard is that they get so much Scripture in all their services, both from the liturgy as well as from Scriptural readings. The average Orthodox Sunday liturgy will have a couple of dozen of verses (whole sections), and sometimes entire chapters of the Scriptures (most often New Testament epistles and the Gospels) read in the service. This is far more than the handful of verses one might get in a typical evangelical service.
I will say that there are signs that this is changing among Orthodox in the U.S., and this is, in my opinion, in large part because of the large number of evangelicals that have been coming into the Orthodox Church in the last 30 years (among some Orthodox groups in the U.S. half the clergy are former Protestants/evangelicals). There are many new resources printed in English for Bible reading and study, much at a non-technical "lay" level.
This is a longish answer, and I apologize, but your question assumed some things that aren't true, so I needed to speak to those, while also answering your question. If there is anything else I can do to clarify what I've written, or to better address your question, let me know.
May God bless you.