Security proof. Adversary source of random coins.
It certainly wouldn't make sense for the challenger to provide the coins. If you are working with non-uniform adversaries, then you can assume that the adversary is deterministic. This is due to the fact that $BPP \subset P/poly$. Intuitively, a non-uniform adversary can receive as "advice" the best random coins that maximize its probability of winning the game. However, if you are working with uniform adversaries, then you cannot assume that the adversary is deterministic.

Note, that as part of the proof, if you need to somehow simulate something for the adversary, then you can always fix its random tape to a uniformly chosen string and then work with an effectively deterministic adversary. However, this is different to having the "challenger" in the game provide the coins; that would make no sense.