My fish are on my hoop stringer which is clipped to my "d" ring on the right side of my BCD. I am left handed so it is usually the left hand that grabs the gills and brings the fish's eye to the shank of the stringer, which is how I open it to avoid any possible mishaps. (I lost a few fish when I was first learning to shoot when it came time to string them.) I know a few people who keep a deliberately weak link in the attachment to the BCD in case it needs to be cut for one reason or another. I do not do this figuring anything bad enough to make me want to cut the stringer free of my person will probably rip the d ring faster than I could get to my knife and begin surgery. A few of the commercial guys I know have been jerked around by sharks that got their stringers without them seeing the approach, stringers that were attached to their BCDs......... and everyone one of them now carries the stringer by hand to keep it in front of them and away from dangling by their legs. I haven't learned that lesson yet.
I personally don't like the idea of leaving the fish unattended or having to return to the anchor line everytime I shoot a fish. Being aggressive works well for me (FYI I'm alittle guy at 5'10" 150, or the same mass as a 7' sandbar, bigger if you add in the longblades, 52" commercial gun and the lp108).
We use lift bags for big AJs and grouper when we are live boating, but thats more for convenience than sharks. 70#AJs and multiple large grouper are hard to swim with, especially in a current. The guys on the East coast of the state by Jupiter use lift bags out of necessity, there are no places more filled with bull sharks with no respect for divers than down there and they have learned to pack hunt in recent years, our bulls and sand bars are not as advanced, socially. If you don't have a boat circling to pick up the bags you have likely just moved the potential shark encounter to the surface where you will b doing your "safety" stop.
My gear is in the pick, although I don't normally carry the lift bag where it is in the pic. FYI Dive Rite makes a great compression sleeve for lift bags to keep them out of the way.