A common rebuttal Protestants give to Catholics when accused of engaging in "Private Interpretation" of Scripture is that Catholics engage in "Private Interpretation" as of Scripture as well, particularly when it comes to a Catholic deciding for themself that the Catholic Church is the church to join. The Protestant envisions that he and the Catholic both are fallibly interpreting Scripture and each are coming to their own fallible conclusions. Given this, the Protestant sees any arguments given for submitting to Rome as not only unnecessary, but even engaging in the same fallible private judgment that Catholics clobber Protestants for doing. In short, the Protestant sees the Catholic engaging in circular reasoning and special pleading.
When I encountered this for the zillionth time, here is the response I gave to one Protestant Blogger (slightly modified for this blog post):
The problem with that claim is there is a misunderstanding (even equivocation) going on with the term "interpret". Really, there are two distinct things going on:(1) Studying the Evidence and coming to a fallible but plausible conclusion.
(2) Authoritatively teaching a binding doctrine, including authoritatively interpreting a text of Scripture.Everyone must engage in category #1. That's not the issue. The issue is category #2. When it comes to addressing category #2, one must see that there either is an authoritative teaching body ("Magisterium") or there is not. If there is no Magisterium, then there are no definitive doctrines, only fallible but plausible opinions. That's basically the state of Protestantism and why fewer and fewer doctrines are seen as "essential". If there is a Magisterium, one must engage in #1 to locate and eventually submit to which Magisterium is the most credible.
Let me give an example of the problem with Protestantism. Let's say that St Paul came down from Heaven into your denomination and told your pastor that your pastor was teaching incorrect doctrines and rather your pastor should be teaching these other doctrines. In the Protestant view, your pastor could theoretically disagree with St Paul if your pastor felt Paul's comments did not align with your pastor's interpretation of Scripture. In the Protestant mind, both your pastor and St Paul were in the category #1 above: they were both fallible men doing their best to discern what the Spirit was telling them through Scripture. Neither could or were teaching authoritatively.
The problem with the above example is obviously that we know St Paul is not on par with your pastor, and in fact St Paul was entrusted by God with the role of #2 above. This means your pastor and his congregation, who are all in category #1, are not free to overturn Paul's teaching should they come to a different interpretation of the Bible. They'd be in the wrong and Paul would be in the right.
What you and other Protestants do is think that a Christian in category #1 has the (optional) duty of locating a denomination and pastor also in category #1. And since everyone is in category #1, then it's possible there could come a time when you disagree with your pastor's fallible but plausible interpretation of Scripture on a doctrine you plausibly but fallibly believe is important, and at that point you could leave to find another denomination or start your own. All the Protestant is doing is shifting between denominations of category #1, completely oblivious to or denying the existence of someone of category #2.
Unless Catholics and Protestants can differentiate and understand these two categories, they will continue to talk past eachother. The good folks at Called to Communion have written extensively about this, but I thought it should be repeated in a more concise form.