There are two orifices under the diaphragm: one round hole on the right side leading to the choke orifice. The oblong hole at the rear leads to an inlet on the top of the bellmouth. When the choke is on, it draws air through the oblong hole in the bellmouth, then on down the round orifice leading to the choke. When the choke is off, the orifice leading to the choke is closed, that is normal operation.
To check the operation of the diaphragm, make sure the choke is off, stick your finger in the inlet and lift the slide, place your thumb over the oblong hole in the bellmouth and let the slide go, it should descend slowly. If it snaps shut, either the diaphragm is leaking, the slide is leaking (doubtful), or the choke is leaking, more than likely it will be the diaphragm.
There are a number of things to check: only the left side carb has a choke plunger, check that, then check the gasket under the choke body. Then check the hose linking the left side choke body to the right, then check the gasket under the right side choke plate.
To check the floats fill a cup with boiling water and immerse the floats, you shouldn't see any bubbles.
Personally, I'd rebuild both carbs and go from there. You'll need a rebuild kit for this. The kits include jets and gaskets, just make sure you blow out all orifices with compressed air after cleaning and before rebuilding the carbs.