Thank you for your response.
I guess I must be doing something wrong, because that doesn’t seem to do anything.
I selected rig1, shift-selected rig2, both in object mode. Both turn a nice bright shade of green. I linked the Animation Data, via CTRL-L.
What should happen now that they share the animation data? When I pose one rig, the other doesn’t change at all. Do the rigs have to be in a special format? Sharing names for the rig, for the bones? Hm… this is a lot deeper into Blender that I have been before.
Copying poses would work, I guess… but I would like to pose both rigs in realtime. So that falls flat.
And another problem I stumbled over: I need the second rig to be mirrored. With a lot of fiddeling, I managed to get that working with bone constraints. But that means I still can pose only the original.
I guess this was just another of my completely crazy ideas that I cannot get to work in the way I imagined.