Good questions...the transfer of air and grasping the out of air diver occurs simultaneously...you grab their BCD/strap with left hand and present air source with right [or vise versa] all in one movement...why not have the out of air diver on the same side as the rescuer's reg hose? Because that is your air and if they panic and rip it out of your mouth the 'fat is in the fire' ...how you control which side the out of air diver is on depends on the rescuer as they grasp them and present air.. you don't predict it you control it after the fact as presented. Could you give up your air and then go to the alternate air supply for yourself? Sure but you still need to control the out of air diver and still stay fully functional yourself.The first move should be to hand the victim a back up second stage (if possible) not grab and try to control the victim. The victim needs air ASAP, anything that delays that process should not be a priority.
Once the diver has taken the regulator, then it makes sense to grab them and take control physically. Man-handling an out of air diver, without first giving them air seems like a non-optimal protocol, since it could easily further their (impending) panic.
I am not sure I understand what the importance of positioning the victim to one side or the other, and I definitely fail to see how this could be predicted or controlled in an emergency.
I've been presented with an emergency air sharing situation (at considerable depth) at least four different times and it can be somewhat chaotic and scary because you don't know how the victim will react.
Tough to consider but the rescuer must control rescue and in my book the rescuer must survive as well as stay operational..
Did not mention another protocol in out of air training...anytime reg comes out of your mouth, you need to blow bubbles, small if needed but exhaling in case of unexpected ascents...whether buoyant or clawing to the surface exhaling, bubbles, must occur. No good to survive out of air only to embolize.
You have encountered 4 emergency out of air divers? Training divers? Wow....great reason to dive solo or only dive with competent divers. My old mantra was: "The ocean only respects competent divers".