It does.
The reason it looks worse for clients is that the bullet lag is severely exxagerating the effect.
In those videos, a guy is aiming about level, and walking forward. Some of the shots go mostly-forward as expected, and the rest hit the ground.
They are not really hitting as downward as the video makes it look, but because the guy is walking forward, and the lag is bad, the player has almost caught up with the shot, if that makes sense.
The host will do the same thing, but since theres 0 lag, the point on the ground where the effect registers is furhter away, and doesn't look as bad.
If you aim up slightly while hip-firing the shotgun (like 10 degrees), it will make the average swing of the shotgun be more centered.