Private interpretation means actually believing what it says!
Not necessarily, however the NT church did not begin under its alternative of the novel and unScriptural premise of ensured perpetual magisterial infallibility as per Rome, but souls discerned both men and writings of God as being so without such, but not as opposed to or without leadership.
Moreover, the very text that Catholics invoke as proscribing private interpretation does not teach that, and thus this use of it by Catholics examples the very thing that they argue against. For that "no prophecy of the scripture is of any private interpretation" (2 Peter 1:20) is not referring to interpretation of Scripture, but that of the writing of prophecy not being the product of man's own understanding, but to the contrary, that the prophets did not know "what, or what manner of time the Spirit of Christ which was in them did signify, when it testified beforehand the sufferings of Christ, and the glory that should follow." - 1 Peter 1:11) Interpret scripture by scripture.