The kind of oil you are using can make a huge difference. Huge difference in smoke, odor, fumes ext. Also what ratio of gas to oil you are using.
If you use a JASO FD/ISO EGD oil, at a 40 to 50:1 ratio, the amount of visible smoke will be so little you wont see it once its warmed up, and even cold they barely smoke, compared to JASO FB oils and other older/non rated oils. Next your chainsaw must be in proper tune, a chainsaw running too rich will very easily give you a headache with excessive fumes and smoke. This is the biggest culprit IMO. Too rich is skipping lots of UNBURNED heated raw fuel in the air, and that can give you headaches/sensitivity in no time flat. Certain brands of oil (even the Jaso FD/iso EGD oils) have different smells and sensitivities depending on the person. Worth trying a few different ones.
Canned fuel here in the U.S.A. is non ethanol and made different than gasoline for a long shelf life, and I have found burns and smells less toxic than regular pump fuel. Next you have 2 basic types of 2 stroke engines in chainsaws. Starto charged, and traditional. Strato charging is a bit more complex but uses something like 25% less gasoline for a given run time. Someone correct me if my number is far off. This will significantly change the amount of annoying fumes. A catalytic converter will reduce fumes (or atleast change their composition to be less annoying), however they don't last forever, and add restriction to the exhaust. Many saws reduce their emmisions so much by using strato charging they don't have a cat. So it might take some looking to find a saw with both. Also don't over look engine size. Its obvious but a 60cc saw is going to put out more fumes than a little 30cc one. Another obvious thing is that if you keep your back to the wind, even the smokiest saw wouldn't matter at all!
For a gas saw your best bet would be:
Smallest CC saw
strato charged
cat converter
Jaso FD rated oil @ 50:1
Aspen/canned fuel
CORRECTLY TUNED!
sharp chain-cut faster/less time cutting
Sure doesn't sound like much fun though!!!!