Go to the sid's Layer Properties - Symbology and check Display Background Value and in the drop down select No Color. I think it likely that the 0,0,0 setting will work but if not repeat with 255,255,255. You can also try setting No Data to No Color.
You may not get a perfectly clean result even after doing this because the sid compression can introduce some data into formerly no data areas, but most of them should be gone. If these are still unacceptable you will have to clip to another image format like a tif which might make for some large files.