Does setting my weapon view in Valve's first-person shooter game Counter-Strike: Source to "left-handed"(in first person it puts your weapon on the left side of the screen) change how other players see me? Will it change how my gun is held in third person mode? Is it only clientside?
No it does not change how others see you.
However, it will change everyone for you.
If you're viewing yourself as left handed, you'll also see everyone else as left handed.
It works the same way for right handedness, everyone else will see you as they see themselves.
The 3rd person character model does not change at all either way.