How does an Intel processor "talk" to an I2C device?
So far I understand, Intel processors use specialized CPU instructions (IN/OUT) to communicate with peripherals. 


Uh, this might have been true in the 1970s and 1980s, but really wouldn't even be very useful for an I²C peripheral, and incredibly intrusive for the programming of the CPU.

So, instead, there's some peripherals attached to a much, much faster bus (e.g. PCIe) that does the I²C talking, upon receiving commands to do so from the CPU; that communication usually happens trough writing data to a specific memory location, instead of using special IO instructions.


  Also, I believe an Intel doesn't have an on-chip I2C controller. 


Oh, I don't think that's true; there should be multiple controllers, but they are purpose-specific, for example, for reading the EEPROMs on RAM modules.


  So, we need an I2C controller, isn't it? 


yep.


  If so, does the processor treat the controller as a peripheral 


yes


  and use IN/OUT CPU instructions to talk to it? 


No, these instructions, as said, are used for a very small set of device types that have been around for a few decades (think: PS/2 keyboards, PCI address space configuration, real-time clock, floppy, parallel printer port, …) only that a CPU needs to directly talk to, mostly for boot and legacy reasons. 

Basically, as soon as you've set up your PCI address space, most peripheral communications are just normal memory accesses.

Often, even, your program would prepare the commands for the I²C controller in a memory region, then tell some DMA controller to transfer that memory to the controller. So, your CPU doesn't even necessarily directly talk to the controller – it tells a DMA controller to tell a say, PCIe controller, to transfer bytes from some memory to a device. 


  Is the controller mounted on the motherboard? What bus connects the CPU and controller?


Depends; it can be integrated in the same IC package, or be on the motherboard.

If directly integrated in the Southbridge (which in Intel CPU chipsets is called the ICH), then the interface would be the Direct Media Interface; it might also sit in a PCIe device attached to the Southbridge, or in a USB device attached to a USB controller, which again can be part of the southbridge, or part of a PCIe-attached USB controller.