I know "her room" is the more natural translation, but I figured it would be "her own room," given that the lesson notes emphasized that this is what makes "sia" different from "ŝia."
The problem is that English and Esperanto have different levels of specificity and different methods of clarification in these kinds of possessives. Sometimes, the word "own" can be added in English to help clarify which of two people (or things) mentioned in the discussion is doing the possessing. But the distinction between "sia" and "ŝia/lia/ilia" doesn't really map to the presence or absence of "own" in English.
In this example sentence, "sia" is required, even though there is no other person in the sentence, with whom we might confuse possession. In English, the "own" of "her own" is always optional, and is generally used to help distinguish between two possible possessors.
Similarly, "Li dancas en ŝia ĉambro" tells us that some female exists, who owns the room, even if we haven't heard of her before. "Sia" can't be used to express this same meaning.
There are three rules for the use of "sia", and related forms like "si, sin, siajn". It's got to relate to the third person (she, he, it , they). The "si" word can't be the subject or part of the subject, nor an adjective describing the subject directly. Finally, it must indicate/reflect back to/relate to the subject of the controlling verb, or verb-like word. Often, it's enough to look at the nearest verb, and figure out its subject, and then use "sia", if the possessor is the subject. If not, use ŝia, lia, ĝia, or ilia. In some complex sentences, it may be a little tougher to pin down the controlling verb, but you will figure this out as you learn the language.
Ŝi dancas en sia ĉambro. Dancas is the controlling verb. Ŝi is the subject of that verb. Sia indicates/relates to the subject. Therefore, the subject possesses the room.
Li dancas en ŝia ĉambro. Dancas is the controlling verb. Li is the subject of that verb. Ŝia does NOT indicate/relate to the subject of the verb, but rather to some other (female) person.