If I recall correctly it runs from the host's system if the host isn't the person who quit, but I don't remember if there is a "new host" selected from the remaining players in CoH on-line when the "first host" quits, for instance in Empire: Total War if the "main host" quits then the player with the best ping (from the remaining players that is) is automatically selected as being the "new host" and it is from that new host's system that the CPU that plays in place of the previous host player who quit is running from.
I'm not entirely certain however, what I'm telling you is what I can remember from reading the modifications section of the official Relic's forums ("The Armoury" sub-forums), and it's been years since I have been there:
http://forums.relicnews.com/forumdisplay.php?f=146
If you're still in doubt (as much as I am) then you could just ask those guys there, they'll surely be able to inform you better on the subject, some of them have done little miracles with modding CoH, they know if from the inside out.