No. A plastic protector will protect the screen from scratches and it provides a tiny bit of shock protection (from hits directly on the screen). A glass protector protects the screen from breaking by taking the shock itself (and breaking) instead of the screen breaking - but it's no harder than the screen, so it scratches just as easily as the screen. A good TPU case, like the Otterbox Defender, protects the phone from shock (I've had mine slammed and dropped) and it has a plastic screen protector - but it is bulky.
The problem with the Note 8, though - and any phone with a curved screen - is that if the curved parts aren't properly annealed, and from the complaints about cracks at the edges, the Note 8 screens haven't been, the curved parts crack just due to the stresses that remain in the glass after manufacture, and you can't protect against that. A screen with alight curves on the edges looks nice, but unless they're willing to put a lot more time into annealing the glass, it's not ready for prime time yet.