I am on plaquenil (hydroxychloroquine) for sjogren's sydeome. What is the drug supposed to do? How do I know it's working

Various. While it may not cure or improve sjogren's significantly, you might expect slowing the progression of your symptoms, such as dryness of your eyes, mouth, throat and other mucosal surfaces (nose, for example).
Unknown. The exact method of action is unknown. It's typically used for fatigue, fever and other systemic symptoms. If it's working then those symptoms should improve.