Newtons laws (which is how we predict motion) apply in an inertial frame. The Earth is a rotating frame, so unless you convert everything to an inertial frame first, you have to account for the motion of the rotating frame (Coriolis, centripetal, and centrifugal accelerations).
For rockets, we usually make the conversion to inertial coordinates and just use straight Newtons laws. But for short range applications like shooting, the rotational effects of the Earth can often be ignored. They are there but third order effects. And the non-engineer laymen likes to think of the Earth frame he is in as an inertial frame for his practical purposes, though its really not.
I have only a little experience out beyond 1000 yards shooting. For anything inside 1000 yards, Coriolis effect is an order of magnitude below other effects, like spin drift, so it can pretty much be ignored. As others have said here, beyond 1000 yards, or when the loft of the projectile keeps it in the air for several seconds or more, then Coriolis might become significant.
That said, for shots that a person (a sniper or long-distance shooter would take), I would consider the CE a much, much smaller effect than factors like wind or gravity.
And if you're a person that wants to be able to make a reasonable effort to hit targets beyond 1000 yards--just to satisfy your curiosity, I suppose you could attempt to apply the Coriolis "rules," but I think they're beyond the comprehension of most normal shooters. If you hit a target at these ranges, it's mostly luck; if you miss, there's more to blame than just the Coriolis Effect.