I'm not an electrician or anyone who knows for sure, but I do have some practical knowledge and common sense (not to say the others answers lack either of those). I do have some experience with electric fences because I've used them to keep cattle & other livestock fenced in for more than 50 years.
Most fence charger's output is intermittent to allow the critter in contact to get away from it once touched. I've seen large animals knocked to their knees from contact with a good 'hot' fence. Chargers have two terminals....a positive & negative. The positive terminal is the hot wire, but it need a good ground for somewhere to discharge....think lightening from cloud to ground. Ground rods are driven into the soil to make better contact with moist soil. If it's really dry, it does help to water the rods to make good contact, because electricity travels much better in wet soil rather than dry soil.
As long as you have a good connections to a good ground rod, I see no reason that you can't run two systems with a common ground rod. It's possible that the design of the charger or chargers could possibly effect the use of two on a common ground, but I have no knowledge of that.
I'm not sure of your question about plugging into different circuits in the house....should make no difference if the same circuit or two different.