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it is hardly a "private interpretation" when the author Himself confirms that He is indeed the author

This is why I was puzzled by the reference to context: St. Paul clearly says (v. 16 and following) "we were eyewitnesses of his greatness". Hence, what is "private" to him is anything that is not succeeding from the apostles, the "holy men of God", who received the inspiration of the Holy Ghost not in ther minds but in observable fact: "Receive ye the Holy Ghost", Jn 20:22.

You are looking for verses 15-17 in:

2 Timothy 3

You must understand this, that in the last days distressing times will come. 2For people will be lovers of themselves, lovers of money, boasters, arrogant, abusive, disobedient to their parents, ungrateful, unholy, 3inhuman, implacable, slanderers, profligates, brutes, haters of good, 4treacherous, reckless, swollen with conceit, lovers of pleasure rather than lovers of God, 5holding to the outward form of godliness but denying its power. Avoid them! 6For among them are those who make their way into households and captivate silly women, overwhelmed by their sins and swayed by all kinds of desires, 7who are always being instructed and can never arrive at a knowledge of the truth. 8As Jannes and Jambres opposed Moses, so these people, of corrupt mind and counterfeit faith, also oppose the truth. 9But they will not make much progress, because, as in the case of those two men,* their folly will become plain to everyone.
10 Now you have observed my teaching, my conduct, my aim in life, my faith, my patience, my love, my steadfastness, 11my persecutions, and my suffering the things that happened to me in Antioch, Iconium, and Lystra. What persecutions I endured! Yet the Lord rescued me from all of them. 12Indeed, all who want to live a godly life in Christ Jesus will be persecuted. 13But wicked people and impostors will go from bad to worse, deceiving others and being deceived. 14But as for you, continue in what you have learned and firmly believed, knowing from whom you learned it, 15and how from childhood you have known the sacred writings that are able to instruct you for salvation through faith in Christ Jesus. 16All scripture is inspired by God and is* useful for teaching, for reproof, for correction, and for training in righteousness, 17so that everyone who belongs to God may be proficient, equipped for every good work.

Indeed, the apostles received the Holy Spirit. But not them alone, nor only them and their successors. Every Christian receives the Holy Spirit:

But ye are not in the flesh, but in the Spirit, if so be that the Spirit of God dwell in you. Now if any man have not the Spirit of Christ, he is none of his. - Romans 8:9

Hereby know we that we dwell in him, and he in us, because he hath given us of his Spirit. - I John 4:13

Moreover, I have testified that I have known the Holy Spirit personally through His indwelling for nearly a half century. This is no small matter. Atheism is laughable to me because I know Him, personally.

Thus for me - and those like me - the "private interpretation" is not the Spirit's authenticating Scriptures and bringing them alive within us - but rather what other mortals try to tell us the Holy Spirit really means by what He said. The indwelling Spirit speaks for Himself.

Howbeit in vain do they worship me, teaching [for] doctrines the commandments of men. - Mark 7:7

Again comparing 2 Peter 1 and 1 Cor 2:

Knowing this first, that no prophecy of the scripture is of any private interpretation. For the prophecy came not in old time by the will of man: but holy men of God spake [as they were] moved by the Holy Ghost. - 2 Peter 1:20-21

Howbeit we speak wisdom among them that are perfect: yet not the wisdom of this world, nor of the princes of this world, that come to nought: But we speak the wisdom of God in a mystery, [even] the hidden [wisdom], which God ordained before the world unto our glory: Which none of the princes of this world knew: for had they known [it], they would not have crucified the Lord of glory.

But as it is written, Eye hath not seen, nor ear heard, neither have entered into the heart of man, the things which God hath prepared for them that love him. But God hath revealed [them] unto us by his Spirit: for the Spirit searcheth all things, yea, the deep things of God. For what man knoweth the things of a man, save the spirit of man which is in him? even so the things of God knoweth no man, but the Spirit of God. Now we have received, not the spirit of the world, but the spirit which is of God; that we might know the things that are freely given to us of God. Which things also we speak, not in the words which man's wisdom teacheth, but which the Holy Ghost teacheth; comparing spiritual things with spiritual.

But the natural man receiveth not the things of the Spirit of God: for they are foolishness unto him: neither can he know [them], because they are spiritually discerned. But he that is spiritual judgeth all things, yet he himself is judged of no man. For who hath known the mind of the Lord, that he may instruct him? But we have the mind of Christ. II Cor 2:6-16

Of course there are many Christians who are more comfortable leaning on trusted spiritual leaders than on the indwelling Spirit. That puts a huge burden on the leaders, trusting them to be following the Spirit and leading faithfully.

I am not here to criticize them for choosing such a path. But it is clearly not the path Chris has chosen for me.