Yes, Android Studio is needed for running Flutter apps on Android but not for iOS. (You don't normally need to launch Android Studio, just have it installed on your machine.)
Similarly, if you want to run on iOS you need a Mac with xCode installed.
(It's rather weird to me that you need to install entire other IDE's as developer dependencies to build mobile apps, but that seems to be how it is.)
> It's rather weird to me that you need to install entire other IDE's as developer dependencies to build mobile apps, but that seems to be how it is.
I don't remember the details, but a couple years ago there was a way to install just the Xcode build tools without installing the entire IDE. I'm sure there's a similar process for the Android SDK. (Though, of course, none of that is officially supported, so you're on your own for the most part.)
Edit: Turns out Google provides downloads for just the SDK without Android Studio[1], so it looks like the separation is unofficial only for Xcode.