First, I would go with different translation in English for the first Russian sentence, smth. like
"Choose your side (in the conflict"
"Choose the side you support"
And the second Russian sentence the best translation would be
"Choose where are you standing"
Basically, the first sentence has a somewhat idiomatic luggage, implying context of conflict, but the second uses verb "стоять" that is mostly used in context of "standing on your legs"
Also, the first sentence should be written as
Выбери, на чьей ты стороне.
It is a complex sentence. It looks weird, because both parts are grammatically incomplete, but there are two of them, and they should be separated by a comma.
But that's nitpicking. As for the actual question.
In Russian connections between words are not regulated by their position, or, more precisely, position is second to inflections. In this particular case, "чьей" is controlled by "стороне" as both words are of female gender and in Preposition case. Usual, i.e. default word order in a sentence is subject-verb-object. The verb in the second part of the first sentence is omitted, so it is subject-object. In part adjective-object, adjective is usually first. So, normally it should be "Выбери, ты на чьей стороне", and it is perfectly fine. However! It is a good idea to specify, what should be chosen, i.e. what links the first part of the sentence and the second. It is "чьей", i.e. what is to be chosen. And so this word is moved forward, and since it does not create ambiguity, it is also perfectly fine. There is not an iron-clad rule for it.