Nonsense. During colonial times privately owned crew served weapons (cannon) and private naval vessels were legally held by those wealthy enough to afford the infrastructure and staffing such tools require. The founders understood that government derives it just powers from those concentually assigned to it by the people. If government has the authority to own Stingers, it comes from individual's rights.
Ultimately, all human systems require trusting the individual actor (soldier, cop or citizen) to do anything, even the right thing. Organized systems may have built in checks and balances to see that power tools aren't abused or toyed with foolishly, but ultimately some individual, or small combination of actors, somewhere has to be trusted.
It's hard to find a cheaper, more common, and more energetic material pound for pound than gasoline, and yet instances of its abuse are statistically non-existent. Stingers may sound more awesome and dangerous (even though their warhead is only about as big as a can of beer), but their potential was obviously less than the tactical deployment of a small team with box cutters and basic piloting skills. It seems pretty clear that not only people in uniform are in danger in the War on Terror, and it also seems pretty clear that if we are going to successfully defend ourselves against such threats, we'll need to trust more than just uniformed public servants.
Nations under siege (Israel comes to mind) invariably count on irregular, non-uniformed citizens to be able to respond. If Hamas had airships, you can bet Israelli citizens would likely demand and develop some form of man portable air defense tool. Then again, Israelli's sense of community hasn't been split apart as badly as ours by diversity, multiculturalism, and the "leave things to the professionals" attitude of a culture in decline.