The verb "is" in verse 20 is the translation of the word 'ginomai' which according to Strong's Lexicon means, "to cause to be, to become, come into being." Hence the sense of this verse is this: "no prophecy of Scripture 'came into being' by any private interpretation." The apostle Peter is here speaking about the process by which the Scriptures came into being, namely, their origin, and not about the understanding of Scripture already given.
Peter says that no Scripture came into being by 'private interpretation' - that is by one's own explanation. Whom does he have in mind? Is it the reader, or the men who penned the Scriptures? Since Peter is speaking about the origin of Scripture, it seems likely that he is talking about the prophets themselves. In other words, Peter is saying that the Scriptures did not originate in the prophets' own understanding. This could be confirmed if we read the following verse since the apostle Peter gives the reason why scripture did not come into being of the prophets' own understanding, "for" he continues, "prophecy never came by the will of man." The prophets did not invent the Scriptures. Rather, they were God's instruments to write his Word: "...holy men of God spoke as they were moved by the Holy Spirit."
So, this verse does not discourage believers from reading and understanding the Bible for ourselves, but it does give us full confidence why we should trust the Scriptures. Though written by men, the Scriptures do not have their origin in the human mind but in the mind of God the Holy Spirit. The Bible is the Word of God.