Comment by HolyBlackCat on Does MinGW-w64 source code need to be cross-compiled in a system that already has GNU utilities?

You likely don't need an existing MinGW installation when building on Linux. But you do need a linux compiler, binutils, etc (so your title doesn't match the question?). You'd only need existing MinGW to build on Windows. As for performance difference, measure it (compile a compiler yourself, compare its performance against a prebuilt one in some distro).