Like said above, Zama carburetors use the "punch bowl" to circulate fuel through the carburetor, not to inject it like an accelerator pump. The purpose of the primer assembly is to circulate new fuel through the carburetor and to eliminate air, aiding in quick starting.
The check-valve on the return spiggot coming out of the fuel bowl area allows air and fuel to return the tank (or on the ground in some models), but does not allow fuel (under pressure from the tank) to flow back through the fuel circuit. The other check valve in the primer assembly blocks reverse flow when pushing the diaphram allowing the pump to operate. I believe there is also a 3rd check valve inside the fuel circuit.
If your return check valve was broken you would flood the saw with excess fuel coming from the tank, but that wouldn't cause fuel to come out of the check valve all the time. Sounds to me like it could be one of the diaphrams, which you would need a carb gasket kit to fix.
A simple light "blow" on the return hose will tell if the check valve is shot, if you can push air past it the check valve is faulty, don't blow too hard or used compressed air. Better to do this off the carburetor.
Can you post a picture of the carb and hoses in current configuration?