Among my close friends, I know Canadian, Israeli, and Australian dual-citizens.
I've never heard of having to "renounce all other allegiances". (Note: it's entirely possible that there is some obscure method of obtaining US citizenship that requires it; but that's not the norm.)
Additionally, my brother became a Canadian citizen (born U.S.), and again had no problems; Canada didn't care about his U.S. citizenship, and the U.S. doesn't care about his Canadian citizenship. He could officially renounce his U.S. citizenship by walking into a consulate and doing so, but there's no up-side to it.
If America would get a draft again, could a dual citizen be drafted?
If that were to happen, there would definitely be an upside to renunciation. ;-)
In general different countries have (very) different sets of rules, see e.g. http://en.wikipedia.org/wiki/Multiple_citizenship
Also if you volunteer (as opposed to being conscripted) for military duty in your non-US nations then the US can kick up a fuss.
But basically if you're a dual citizen with a nation friendly to the US then you really have nothing to worry about.