How realistic is DIY sharpening of scissors on a stone?
Like all projects a certain amount of skill and the proper tools are needed to sharpen scissors correctly.

A 120 stone is not fine enough to put a fine edge on a pair of scissors. Something a lot finer would be needed.

Many knifes are sharpened on both sides of the blade. Normal scissors are sharpened on the angled side of the blade. Some scissors can be disassembled to make sharpening easier, others cannot.

If your scissors are valuable, take them to a sharpening service. They should come back good as new.

If your scissors are not worth paying to have them sharpened, and are too dull to use, try sharpening them yourself. You will learn how well a 120 stone works for this task. You may find you have an interest in sharpening things. Or you may find you need to buy a new pair of scissors.

If you do try sharpening them yourself, be careful. Even dull scissors can cut your fingers.