Messages that you receive and send are transmitted through Instagram’s network and are not retained on their systems. Whether it’s comments or direct messages, they’re only saved on your smartphone/personal device and nowhere else.
However, Instagram does store your photos and may use them for its own reasons. Since Instagram usually profits from them, they are subject to a different treatment. That is why, when you sign up for the network, you basically withdraw your potential copyright claims, and hence they are treated differently than messages.