Yumemitai:I've just started learning Esperanto, and I came across the use of these words as conjunctions.
However, if word order is mostly free, and these question words can be used as conjunctions, isn't it ambiguous which one is the question and which one is the conjunction?
For example:
"Kie estas vi, kiam ŝi estis mortigata?"

You can exchange the two parts of the sentence:

"Kie estis vi, kiam ŝi estis mortigata?" = "Kiam ŝi estis mortigata, kie estis vi?"
Both forms are questions.

"Mi estis en Nov-Jorko, kiam ŝi estis mortigata." = "Kiam ŝi estis mortigata, mi estis en Nov-Jorko."
Both forms are not questions.

And RiotNrrd is totally right,

"Kiam ŝi estis mortigata, mi estis en Nov-Jorko."

is a short form of:

"Kiam ŝi estis mortigata, tiam mi estis en Nov-Jorko."

Yumemitai:
However, if word order is mostly free, and these question words can be used as conjunctions, isn't it ambiguous which one is the question and which one is the conjunction?

Interesting, I think I now understand your question. And indeed, there is no difference in the words between:
"Kie vi estis, kiam ŝi mortis?"
and:
"Kie vi estis?" "Kiam ŝi mortis?"

In written form you of course have the question marks. I never thought about that before.

I don't think people would generally run the two sentences into one sentence in a way that is unclear. I think in speech, you would either get "questioning" intonations at the appropriate times (assuming the speaker comes from a culture that does that), or at least a pause between the two questions to make clear that they are separate items, if they indeed are. While it's totally possible to run them together or split them, and that alone changes the meaning, speakers are (hopefully) generally aware of any ambiguities in their constructions and I would expect would compensate for them.

That doesn't mean that no one would ever say something ambiguous. People say what they say. But I don't think this is something that is a particular problem in Esperanto. More of an "edge case" kind of issue that is typically solved by context or intonation or some other obvious means in speech, and (as mentioned) by question marks in text.

I think Yumemitai isn’t asking how to tell two separate questions from one, but how to tell which of the two clauses is the main one and which is the subordinate.

Yumemitai:However, if word order is mostly free, and these question words can be used as conjunctions, isn't it ambiguous which one is the question and which one is the conjunction?
For example:
"Kie estas vi, kiam ŝi estis mortigata?"
If I didn't make a mistake, this means "where were you, when she was being killed?"
However, because the sentence order could be either way, this could also mean, "Where you were, when was she being killed?" (Which, I suppose, could imply that you live in a place with a different time zone, so while she was being killed, it was a different time for you.)

The only solution is either picking up which is which by context, or having a particular word order be correct.

Or, there's something I don't know of yet. How does this work?

Unfortunately, as far as I know, written Esperanto has no way to disambiguate such a question. In speech, intonation would usually make it clear. Borrowing punctuation from Spanish to delimit the part of the sentence with interrogative intonation:

• ¿Kie estis vi, kiam ŝi estis mortigata? “¿Where were you when she was killed?”.
  
• Kie estis vi, ¿kiam ŝi estis mortigata? “Where you were, ¿when was she killed?”.

So, barring innovative punctuation, in written Esperanto you probably should find a different wording.

Yumemitai:
The only solution is either picking up which is which by context, or having a particular word order be correct.

Or, there's something I don't know of yet. How does this work?

A solution needs a problem, and in this case I must say I struggle to see what the issue is. Despite its apparent ambiguity the phrase "Kie estis vi, kiam ŝi estis mortigata?" would always be understood to mean 'Where were you when she was being killed?' If you want emphasize the place then use tie in the way novatago suggested: Ĉu tie, kie vi estis, kiam ŝi estis mortigata?

The problem with the example sentence is that a person is generally only killed once and in one place. However one can construct similar sentences:

Kie vi estis tiam, kiam ekpluvis? = Where were you at the time the rain began to fall?
Kiam ekpluvis tie, kie vi estis? = When did it start to rain at your place?

If we now omit the "tiam" and the "tie" we actually do have a little ambiguity, but --as novatago pointed out-- this can be easily avoided by not omitting these words.