How do Christians who see Scripture as self-evidently inspired explain that Catholics, Protestants and Mormons have different sacred scriptures?

John Owen summarizes well my belief concerning the nature of scripture:

I affirm, That it is the work of the Holy Spirit to enable us to believe the Scripture to be the word of God, or the supernatural, immediate revelation of his mind unto us, and infallibly to evidence it unto our minds, so as that we may spiritually and savingly acquiesce therein. (Owen, J. (n.d.). The works of John Owen. (W. H. Goold, Ed.) (Vol. 4, p. 15). Edinburgh: T&T Clark.)

First, It makes sense that the Scripture must have its authority from its own internal self and therefore be ‘self evident’ – not needing men to authorize or prove its authority. Men are useful in confirming that evidence but the original source of faith must not come from men, and therefore not from reason or logic itself, as the origin of the authority of scripture to the mind.

Second, being self evident would not imply everyone can see it – due to sin. It is collecting guilt not to believe self-evident truths from God:

18 The wrath of God is being revealed from heaven against all the godlessness and wickedness of people, who suppress the truth by their wickedness, 19 since what may be known about God is plain to them, because God has made it plain to them. 20 For since the creation of the world God’s invisible qualities—his eternal power and divine nature—have been clearly seen, being understood from what has been made, so that people are without excuse. (The New International Version. (2011). (Ro 1:18–20))

Third, as the scriptures are infallible and spiritual it is the Holy Spirit alone that illuminates the mind in order for one too see the self-evincing truths within them. Those who believe remain fallible and may differ to what degree they accept the infallible truths illuminate upon their minds. Even as individuals we will always grow and learn and so may spend our lives finding more and more of those infallible truths in the scriptures as they are an endless universe of faith, hope and love.

18 And we all, who with unveiled faces contemplate a the Lord’s glory, are being transformed into his image with ever-increasing glory, which comes from the Lord, who is the Spirit. (NIV (2 Co 3:18).)

From this mountain peak of illuminating truth, to descend into the carnal and dirty divisions between Christian denominations who claim a different cannon of scripture: the answer is some have more genuine faith and resist the Spirits illumination less than others. And some have no faith at all.

I can't speak for Catholics and Protestants, but as a Latter Day Saint, our view of scripture might be a bit different than others. Scriptures to us may not be as 'untouchable' as to others. To us, scripture is valued because it can contain the word of God. Though, we recognize that the words we read on the bible's pages have been through quite a lot. First, concepts or ideas were conveyed into the mind of a mortal prophet (most of the time). Then those concepts were articulated into words or writing; Then they underwent generations of opaque hand-copies and translations. Then they underwent tests of authenticity, consistency, and quality to be deemed as 'canon' by the catholic church in the first few centuries A.D. A lot can be lost in those processes, and indeed we believe a lot was lost. Though we treasure what remnants we have of prophetic utterance in previous dispensations, we in no way believe it is a complete collection of all prophetic revelation.

With that said, we believe that the gift of discernment is a gift of the spirit, and can communicate to you the truth (or otherwise) of anything that is claimed to have come from a prophet of God. A person can receive this gift through prayer and faithfulness. Someone filled with the spirit will likely not only understand at a deeper level the scriptures they are reading, but will also exclaim that the scriptures are self-evidently true. Someone without it will struggle to know if what they're reading is true. As with all gifts of the spirit, they are received by asking and faithfulness. They remain with us conditionally upon our faithfulness, and can be withdrawn at any time.

Articles of Faith:

8 We believe the Bible to be the word of God as far as it is translated correctly; we also believe the Book of Mormon to be the word of God.

As for the distinction between biblical scripture and Book of Mormon scripture. The Book of Mormon is similar to the bible in that it contains records of many prophets' writings as well as a nation's history. At its root, it is a collection of prophetic revelation. A key difference is that it was compiled by an authorized prophet and historian named Mormon around 400 AD, who was given visions of our day, and guided as to how to choose what to include. He exclaims that he had couldn't include even a hundredth part of what he would have liked to, so what we do have is deemed by him to be the most important parts. Also, the translation of this record into english was a revelatory process by Joseph Smith. So in that light, we consider the words of the Book of Mormon to have more authenticity and weight because they were guided by an authorized prophet and revelation at each stage.

4 And when ye shall receive these things, I would exhort you that ye would ask God, the Eternal Father, in the name of Christ, if these things are not true; and if ye shall ask with a sincere heart, with real intent, having faith in Christ, he will manifest the truth of it unto you, by the power of the Holy Ghost. 5 And by the power of the Holy Ghost ye may know the truth of all things. (Book of Mormon: Moroni 10: 4 -5)