I don't know of anyone using stabilizer for hand embroidery, but it is a necessity if doing it on an emb machine, as the thread tension will pull the design so it puckers. If you choose to embroider using batting behind, either way, you don't need stabilizer.
Sometimes a backing (stablizer) is suggested to help hide threads that may cross over or been seen from the front. If I do this, I generally trim it very close to the stitches after completing.