The Roman Catholic canon was defined ecumenically at the Council of Trent (19th century). It is basically a final approval of the canon that the Church of Rome had always been using.
The Eastern Orthodox canon was defined ecumenically at the Quinisext Council (part of the 6th ecumenical council). More precisely, the council gives ecumenical authority to the canons of several saints (Cyril of Jerusalem, Hilary of Poitiers, Athanasius of Alexandria, Amphilochius of Iconium, Epiphanius of Salamis, Basil the Great, Gregory of Nazianzus) and local councils (Laodicea (363), Carthage (419)), and the Apostolic Canons (minus 1 and 2 Clement and the Apostolic Constitutions). The current Eastern Orthodox canon is simply all these books put together in the same list (because these lists aren't identical to each other). On top of that, you have 4 Maccabees in the appendix of the Greek Bible and 2 Esdras in the appendix of the Russian Bible, but they are not considered to be inspired.
The Oriental Orthodox… it's more complicated. They haven't had an ecumenical council since the 3rd one, which means that they haven't had an occasion to give ecumenical authority to any one Biblical canon. So each church has its own traditional canon, and even within that church they are still in the process of defining what is and isn't inspired (for instance, Jubilees and Enoch are extremely popular in the Ethiopian church, yet the synod technically does not list them as being canonical).
>How is a Orthodox Christian supposed to know if they're using the correct Bible?
The Bible isn't a single book, but a library of books. A better wording for your question would be "how is an Orthodox Christian supposed to know if the texts they're reading are all inspired?". The answer to that is 1) faith that our tradition is moved by the Holy Spirit, and 2) an inner movement of the Spirit that lets us know that our everyday Christian faith is the one that is depicted in those texts, and 3) with prayerful discernment we can see that these texts are about Jesus Christ and are doctrinally in accordance with the gospels.
But, it's not a sin to not read all the inspired texts. There are several books of the Bible that aren't read liturgically at all, in fact. Most notably, we don't read from Revelation.