if you got modular hand and you put propulsion system and some pilot yep afik
Not quite what I had in mind. More having a left arm that can go into drone mode while still attached to the body, firing weapons from it's own dice pool and using it's own initiative, etc.
For starters, you would have to pay for the node that you would be installing in the limb. Whether it's a commlink, an autonomous drone system, or whatever, those System, Response, Body and other rating points aren't free. And you would have to pay for them instead of trying to argue that the regular cyberlimb has to have default values because none of that functionality is needed in, and would actually be detrimental to, the regular functioning of a cyberlimb. (And I would expect that you're looking at a custom pilot program, especially if you've done anything cute to your cyberlimb, just as an excuse to make you pay extra for it. "You want a pilot program for a what?!?! I heard a story about a crazy guy that did that once..."
)
Then you're going to need to arrange for sensors for that cyberlimb. You'll get touch and temperature for free, but touch won't do you much good for firing a gun.
And, finally, that gun's going to think you're a huge jerk for constantly spoiling its shots, and you're going to hate it for occasionally blocking yours. One person in control of both of his arms can coordinate which way he's facing to shoot two different targets. If your left arm was moving on its own choosing targets, it's not going to be coordinating with you about which way you need to be facing. (Do you want to pay extra for your own personal TacNet so that you and your arm stay coordinated? It would be a start.) I would say that negative modifiers for things like recoil would bleed over from you to your arm, and vice versa.
Of course, you'd still get the dice pool modifiers to the character itself for losing the use of the cyberlimb while the drone was in control of it.
That's pretty much just my attempt to apply common sense and the rules to the situation.