What would cause a chainsaw to only start when trigger is pulled?
Some saw manufacturers, when starting is difficult, specifically direct the user to lock the throttle "On" to start, then depress the throttle immediately, to bring it to idle. If you do mess with the idle screw, make sure that you do not increase the idle speed such that the clutch drives the chain to move when idling. 

You might try "flashing" the choke a couple times while the saw is running (choke the engine for a brief moment) as this helps clear air from the fuel system (The choke restricts airflow, creating a higher vacuum pressure that pulls air out of the fuel system.)

If you have messed with the carburetor, make sure you follow the user manual. Most have 3 adjustments: Low Speed fuel/air mixture, High Speed fuel/air mixture, and Idle Speed.