Whether the BG has the weapon in his hand or in his waistband, I think the best action would be to "secure the crime scene" and make sure that the BG does not have access to any weapon in the event that he is alive and decides to shoot back or run and dispose of his evidence before going to the hospital. Alternately, what if a witness decides HE wants that shiny gun laying on the sidewalk near the BG's hand and picks it up and runs with it. You cannot leave the crime scene. I can just see it now, "NO really officer, he had a gun but someone else took it."
So, I would take the BG's weapon and then provide it to the police when the arrive, which is also a very delicate matter. No sudden moves, comply 100%, etc.
I would NOT steal it, for reasons that others have offered (ethics, could tie you to a prior crime, could be stolen, etc.). After a shooting you can be assured you will be, at the minimum, completely searched and placed into a patrol car and taken in for questioning. They'll find the gun at that time and you're credibility will start to drop fast.