Does Sola Scriptura go hand in hand with personal interpretation?

First off, your basic premise is spot on. As a lifelong evangelical, my biggest gripe with evangelicalism is that we either dismiss or declare heretical anything that is based on "Tradition," ignoring the fact that all readers of Scripture do so with a tradition in mind. Any theological framework (rapture? trinity? the words don't appear in Scripture!) frames the way that a passage is read, which in turn heavily affects the outcome.

As an example, Isaiah 7:14 is clearly read as a prophecy of Jesus - and the Gospels reinforce that it should. And yet, the context clearly states that it was supposed to be fulfilled immediately. But, because of tradition, even the Gospel writers also "knew" this was supposed to speak of the Messiah.

As another example, every time a true "Fire and Brimstone" sermon is preached, the fires of hell are quite warm. And yet, the biblical warrant of Hell is limited at best. (I'm not saying Hell doesn't exist - quite the contrary - but find it in your Bible, you may be surprised how little the Bible talks about it.)

So, in both of these instances, I suggest that the "personal interpretation" is, regardless of what the person may want, heavily guided by a tradition. What differs are two main points:

1. The Protestant explicitly rejects the Magisterium.

While they don't voice this, to many Protestants, the Pope is still selling indulgences and banning people from reading the Bible for themselves. This is patently false, but it informs the worldview.

As such, the typical Protestant exercise in interpreting scripture is to try to analyze it in a vacuum, trying to get back to "the first church," (never mind that even the first churches had their own problems - see Corinth, Ephesus - Rev 2, etc...) This tradition causes the believer to ask "What would this have meant if X had not been the historical interpretation" which colors the analysis.

1. The doctrine of the priesthood of all believers was promulgated precisely to fight against this excess. Traditionally, Protestants think "the priesthood of all believers" means that there is no need for a priest. Go back to the original doctrine, however, and you'll see that what was meant was that every believer would serve as a priest to everyone else. Private Interpretation was regularly denounced by Calvin and Luther, precisely because of the bad interpretation that could result. The priesthood was supposed to be the corrective precisely to private interpretation.

In the end, however, that hasn't worked out terribly well. The marriage of philosophy and theology and the opening of seminaries to a much wider audience has been the new tradition in which Scripture is read. It has become traditional to refer to theologians like Ryrie, Scofield, Lewis, LaHaye, etc... instead of the Magisterium, but in the end, we've really gotten back to just having a separate tradition that we cite.

A sizeable number of Protestants would argue that the Left Behind books are built on the exact timeline of the end times "based on Revelations." Mind you, they can't get more specific than that (because it ain't there folks!), but we have that timeline in our heads and accept it as dogma for all practical purposes.

We'd also say that the Four Spiritual Laws are effectively dogma. Each point in the 4 Spritual Laws has scriptural warrant, yes - but the notion that "this is what it means to be saved" is tradition, not scripture.

To answer your question another way, no you can't define something as dogma strictly by tradition, but you can effectively arrive at the same result.