Well, exactly as much as any other cpus can do by solely controlling it's output electrons: not much. Let's not ascribe deity-like powers to things that we can understand fairly well.
You're again using terms incorrectly. A "side channel" implies that someone is listening to information that is unintentionally leaked. Unless your expectation is that this CPU is going to start side channeling our minds with the EM waves its emitting (which again, "deity-like attributes"), we'd need to be specifically listening to whatever "side channel" it uses, and it would require knowledge of and access to that side channel.
Something being able to send additional information over a side channel doesn't help unless that information is received, and so realistically, unless your hypothesis is "mind control/hacking the airwaves/whatever via sound waves the chip emanates" or similar, which are preposterous, it'll always be just as easy for the thing to transmit information via the normal channels.
The thing about all of these is that they generally allow you to get a small amount of data out that can sometimes help you with things. But again, without ascribing magic powers to the system, all the stuff that it can directly affect: power draw, temperature, disk spin speeds, monitoring LED blink speeds, noises, even the relatively insane things like EM frequency emissions can all be controlled relatively easily, and no matter how smart it is, I don't see an AGI violating physics.