If you control for the demographic and economic factors, which both Lott and Donohue do, and CCW really reduces crime, a statistically significant relationship should still be apparent.
The only way I can think of that you might not detect a true relationship is if it is nonlinear and you use a linear specification. Suppose, for example, that CCW significantly lowers crime if it is already high, but that its effect on crime diminishes as the crime rate goes down. Since crime rates were lower during the 1993-1999 period for reasons mostly unrelated to CCW, the researcher would not find any beneficial effect of CCW, even if he controlled for demographic and economic factors.
Of course, when you start talking about non-linear specifications, you open up a whole new can of worms.
I would believe that CCW would have exactly such an effect.
I think common sense demands that it not be linear, specifically that it has more effect in high crime areas. Moreover, common sense is the most reliable guide in studying such trends. The vanity of Socialogy as a hard science that should be approached in the same show me manner as Physics or Chemestry is very annoying to reasonable people. No tightly controled repeatable experiments are involved. So the Donohue study boils down to "you can't be absolutely sure about everything Lott did". This is true, but at least Lott had some common sense!
To spell out the obvious: Concealed weapons might stop a little crime by being shot, more by being displayed, but stop the most by pontentialy existing. Thugs really really hate it when their victims might be armed. Thugs are numerous in areas high crime areas. Thus one would expect this third and most potent effect to be higher in these areas.
Left wing psychosis does not even allow for the consideration of certain obvious concepts someone being though of as a "thug". Missing the obvious in forming initial premisis for such an analysis makes it complete hogwash. Such hogwash gets Left wing faculty tenure, prestige, and influence. Resonable people are offended by this.