There is a frequent charge by Protestants that whenever a Catholic presents Scriptural evidence for any particular doctrine, rather than engage the passage in question and interpretation of it, the Protestant simply responds by saying something to the effect, "is that your private interpretation, or did the Magisterium infallibly interpret this verse for you?" The intent is to neutralize the Catholic argument by re-directing the issue onto that of infallibility.
This Protestant "response" is simply misguided. It fails to distinguish and understand the Catholic approach to Scripture and the Catholic objection to the Protestant error known commonly as "private interpretation." The truth is, both Catholics and Protestants are guided by a teaching traditionally called the Analogy of Faith (the classical meaning of the term 'analogy' is 'proportion' or 'rule'). The Analogy of Faith is the teaching that all theological interpretation must be done in harmony with the Church's defined doctrines. This means that the Church doesn't need to infallibly interpret every single passage of Scripture, but rather only needs to set up certain 'parameters' (i.e. dogmas) from which to read Scripture in light of. One of the preeminent examples - which both Catholics and Protestants would readily agree upon - of the Analogy being used is the quote from St John's Gospel where Jesus says, "The Father is greater than I." Knowing that the Church teaches Christ is a Divine Person with a Divine and human nature, the Analogy of Faith tells us that we cannot interpret Jesus' saying in such a way as to contradict that dogma.
The point of real divergence is that Protestants don't have a definite way of establishing dogma, where as the Catholics do (via the Magisterium). The result is that Protestants have less definite 'parameters' to operate within, leading to widely divergent interpretations of Scripture. That said, the Protestant Dogma of Sola Fide (Justification by Faith Alone) is the most important unique 'parameter' when it comes to Protestants interpreting Scripture. A good example of a Protestant using their own Analogy is when they approach James 2:24, in which the Protestant knows they must interpret this passage so as to not contradict Sola Fide.
Back to the original point: when a Catholic presents a text of Scripture to demonstrate or prove a given concept or teaching, they need not have a specific Magisterial interpretation of that verse, since the Church allows doctrines to be demonstrated or defended in any legitimate manner - particularly by applying the Analogy of Faith.
The issue of infallibility (i.e. authoritative interpretation) comes into play when defining the dogmas in the first place. A classic example of the need for authoritative interpretation (aka infallibility) is when examining the Biblical phrase "This is My Body." While the phrase is perfectly intelligible, what is not "clear" is whether this is to be taken literally, figuratively, or somewhere in-between. While many modern day Protestants would say the actual interpretation is ultimately "non-essential" to salvation and thus one is free to hold any view, historically Protestants have had bitter disputes over how to view this saying. Generally, the Anglicans and Lutherans hold to a mostly literal view, the Calvinists hold to a view between literal and figurative, and the Baptists hold to a mostly figurative view. Since none of these views directly contradict Sola Fide, the Analogy doesn't help one way or the other, and ultimately an authoritative interpretation (by a Magisterium) is necessary to settle the dispute.
At this point, only the Catholic position makes sense, since the Catholic position openly affirms the existence and necessity of a Magisterium. If Sola Scriptura were true, then Scripture alone would have been clear enough to settle the dispute. As it stands, Scripture is not formally sufficient, as the "This is My Body" example plainly demonstrates.