Mine also run off the house battery. I do have a house battery disconnect, but the steps are connected on the always-on side of the disconnect, so they can still be used. (For that matter, the converter is also connected to the battery side of the disconnect switch, but that's atypical.) My steps aren't automatic, so they don't actually use any power unless the control switch is pushed one way or the other.
I suspect the chassis battery connection may be somewhat more common, particularly with the systems that have automatic retraction when shifting into gear. It's easier to wire up the controls and so forth when you don't have to worry about separate electrical systems, which may or may not be active or tied together at any given time.
At any rate, they're a 12V system, so they're not directly powered by shore power when plugged in. They may be indirectly powered via the converter (if connected to the house batteries) or, much less commonly, via some sort of a chassis battery charging system (if connected to the chassis battery and there's some sort of a charger for it on the rig).
