Yes, but you have to make sure that the objects (rigs) are identical.
The Action Editor
Either open up a Dope Sheet window or change to the Animation Screen layout.
Change to the Action Editor in the dropdown menu where it likely currently says Dope Sheet.
If you select the rig that you already have animated, it will probably display something like ArmatureAction where it says + New in the above screenshot. If you select the unanimated armature, you can select ArmatureAction from the dropdown menu to the left of + New.
This can work for any number and any type of object. For example, if I have a cube keyframed for X-Location and X-Rotation, I can use the same method to copy its movements to a nearby cone and cylinder.
The demo .blend:
Hope this helps; feel free to ask questions!