Any of the varied gauges will work for HD in trained and capable hands.
The criteria for a HD shotgun are:
It must be reliable.
It must be simple and fast to operate in times of crises.
The stock must fit the operator.
It must hold at least two shots.
There's a few other things not quite as crucial, like being short for handling in tight places. These tend to be self evident.
Many shotgunners,including myself, choose repeaters for HD. Southpaws seem to not be terribly bothered by using right handed guns for the most part. Those that are can use the Ithaca 37, the lefthanded 870 or the BPS.
Try out a few shotguns and see what works for you. While you're at it, get off on the right foot and take a lesson or two.
The Archives have tons of stuff on HD shotguns, research there will yield pertinent material by the gross.
HTH...