You should check Mirroring.
If you want to mirror your animation form the left side to the other
side during pose mode you just need to select the whole left side then
copy the pose with Ctrl+C.
After that press Ctrl+i to go to your right side
and finally press Shift+Ctrl+V to
paste the the X flipped pose, that way you'll have a perfect mirror ...
Since Blender 2.81, a new feature has been added in order to overcome the limitation of one same rig per scene: library overrides.
It's only the first implementation and officially considered as experimental, it is still limited and it's planned to be enhanced in the future.
You can learn more on the manual