This is okay. The straps are adjustable so you really don't NEED to screw the brackets in the top and bottom. And in fact, if you want screws holding the bottom brackets in place, you're going to have to take the door off it's hinges. I went ahead and put the screws in the top brackets, and my one complaint is that both screws went half way in and then stripped, even with the full length guide hole I drilled for them. The screws are made of weak material. I ended up hammering them in the rest of the way and it seems like it will hold, but I'm going to have to cut a new notch in the screw head if I ever want them out.
As a side note, I specifically bought these for hats and light items, and they are plenty strong enough for that. Whatever you're planning on putting on here, it needs to be relatively light. Think corded earbuds, necklaces, neck ties... Otherwise it might bend and fall off after a while.