Some (many) years ago when the GCRR was running steam in regular service during the summer a friend and I chased the comings and goings in Coconino Canyon. I seem to remember it being very accessible and do not remember any no trespassing signs or anything like that. The canyon is in Kaibab NF so it is all public land. We did a lot of walking along the tracks looking for good spots. I think hiking is fine, not sure about ATV's. One thing we found interesting was/is the back roads provide a "free" way to get in and out of the Park. Here are a couple of pix. You will notice the regular trains do not provide the nice smoke you get on a charter runby.
JBWX