Honest question - which games had server side hit detection? I genuinely can't think of any that I've played which did, and a quick Google suggests that Counterstrike (both Source and 1.6, the earliest online FPS games I remember playing and which both had dedicated serves) for example had client side hit detection.

Intuitively it feels to me like server side detection is the worst of all worlds because it wouldn't definitively solve the problem of being shot behind cover (as you won't necessarily be behind cover on the server) but would add something in the region of 40-60ms lag (on a good server!) while you wait to get feedback on your bullets registering.