I've a pretty common machine, a few years old, but still quite capable with regards to my needs: i7 3770, 16GB RAM, RX470 8GB, 2 monitors
I'm planning to switch (dual-boot with Winblows for a while) to Linux for everything I do.
The problem is, I do game on my computer.
The first thing I tried was the Windows VM with the discrete GPU passed-through. That didn't work (strange Code 12 error in device manager) and I abandoned the idea as it would have meant that I couldn't use both monitors in Linux (without rebooting).

So here I am, asking advice on how to drive the monitors. Do I connect both to the discrete GPU and disable the iGPU? Or do I connect one to the iGPU and the one that I game on, to the discrete GPU?
Is it wise enabling both GPUs? What I'm worried about, is possible incompatibilities between GPUs and their capabilities? How will X use them? If I run both GPUs, which one will decode a video for example, while I have a game running? Will it be the one it is displayed on, or the one with the most complete decoding capabilities? Same for a browser with hardware acceleration, will it slow down the game or run on the iGPU?

I'm not new to Linux, just to running it as a desktop.
Any thoughts are appreciated, pros-cons, previous experiences, etc.

This probably is the best way to go, if you can make it work. It will let the Windows VM use the graphics card to the best effect, while the Linux host can rely on the iGPU.

UnRAID is proven for multiheaded gaming rigs, they have a graphics card per station along with a strong CPU and lots of RAM. The technique may be adaptable to your use.

Some virtualisation software (such as VirtualBox) offers 3D acceleration, but I doubt performance will be good.

If your monitors have multiple inputs, connect one to the iGPU and one to the graphics card and use the monitor controls to switch inputs as desired. If they don't have suitable inputs, then either physically plug and unplug cables or get a KVM.

@cantab

I shouldn't have to push any buttons or fiddle with cables just to have the windows I want on the display I want them on, in this age.
I understand your reasoning though, the best gaming experience is on Windows. There are a lot of hoops to jump through to get the games to run on Linux, and a good portion of them won't run at all (for various reasons, like DRM).
I am committed to the Linux path though. What will run, will run. For what won't, I'll other restart into winblows, or I'll stop playing altogether.

My question was: what happens in X.org when you have 2 GPUs, each with a monitor hooked up, but with very different capabilities? Can you choose which GPU renders each window / accelerate each application?

Traditionally it has been pretty routine for the motherboard BIOS to limit GPU access to either the IGP or the PEG. This seems less common now than it used to be. With a more permissive BIOS you can designate which GPU and thus display should be primary or preferred and use both. It's mainly up to software how to use the available displays. With Xorg the default configuration is for there to be one extended desktop, called "screen", with each display utilized at its preferred mode. Usually this default places the primary display to the left of the slave, but reversing sides or over/under are readily possible.

When the difference between display capabilities boils down to pixel density, things can be difficult. Xorg allows use of only one logical density at a time, so object sizes between displays can vary from none at all to painfully disparate.

Choosing windows or screens on a per app basis I've never attempted.

@ mrmazda

I ended up disabling the iGPU and driving both monitors from the AMD GPU. I figured there would be fewer things that could confuse Xorg, even though having a bunch of websites open on the screen I'm not gaming on might slow down the games.