God inspired the authors of the Bible. The Bible was written by "their" inspiration they recieved from their God. Years later, the Church decided which Books were God inspired and which were not. We have gone by these Books for centuries.
Who has the authority to decide (in both the Torah and the Christian Bible) what books are inspired by God?
Without pretending to demonstrate in some perfectly neutral and objective way the "correctness" of this approach, which I don't think is possible, the eastern orthodox view is a little different from the way questions about biblical authority and inspiration are usually presented, and I think somewhat more tenable, so I'd like to mention it because I think it might be interesting.
As far as authority, eastern orthodoxy doesn't view it in terms of ascertaining that some books are "inspired" and some are not, such that inspiration is the criteria determining what is scripture. Instead, the authority that determines a canon of scripture is that of the Christian church as a whole, historically exercised through ecumenical councils, although the means are secondary, a practical matter. Ideally, the church as a whole, as a body, has the authority to select what books it recognizes as scripture for itself, because the church predates the texts. In the early Christian church, the criteria the church used to determine books of scripture, speaking generally and ignoring a lot of details, was that of apostolic authority. In other words, they believed that certain texts were more closely associated with apostles, or held more closely with the views of the apostles, and authority derived from that source, because the apostles were appointed by Christ. We know now (or at least greatly suspect) that some of the "apostolic authors" of texts were really pseudopigraphic, but it's still useful to understand the means that the early church used, and allowing for some fuzziness between apostolic authorship and merely apostolic tradition, it's at least coherent.
As far as "inspiration", as Christians used it, it comes from 2 Timothy, and as you said, eastern orthodoxy views the authors of biblical texts as being inspired, that is: filled with the Spirit. Your phrasing, which distinguishes between an
author being inspired rather than a text, is important, I think, because it makes it easier to be more nuanced about questions of "infallibility" and error and all of that in a reasonable way. In any case, from the eastern orthodox perspective, the church may recognize inspiration in many lives (c.f. the lives and writings of saints), but it does not decide in some final way what is inspired, and inspiration was not the criteria for choosing a canon.