Black fungus and wood ears mushrooms are not the same although wood ears are often substituted for black fungus. Both are often sold dried and they need to be rehydrated prior to cutting and cooking.
I was lucky to find fresh black fungus (or, perhaps, they are wood ears — I can’t tell just be looking) recently, rather large, which were perfect to illustrate which the inedible woody parts are and how they should be removed.
See where the white arrow points? If you feel around with your fingers in that part, you’ll know how much different the texture is from the rest of the fungus. That tough hard part should be cut off.
Take a small knife and cut around the tough portion of the black fungus, making sure that only the jelly-like portions remain.
After the trimming the black fungus, you can cut them into whatever size you prefer. Roughly chop them, thinly slice them, finely chop them… whatever the recipe calls for.