Whether your sister or you can alone alone or must act together depends on the wording of the POA and whether you're referring to a Durable POA or a medical one (Living Will). The terms of anything that creates joint authority should specify whether either of the individuals granted authority can act independent of each other or must act jointly.
So you'll need to check the document to see how authority is created.
Time for you to have a reality check with sis. It's not an easy decision for anyone to move mom to a NH. It would require a doctor approval and the NH would also evaluate mom to see if it is a suitable placement for her. It is not something decided by one person. Try to get all the details.
So you'll need to check the document to see how authority is created.