Yes, internal mechanism(s) selectively activate the ejector on fired barrels only/
When the trigger's pulled on a barrel & it's hammer drops, the cocking mechanism for that hammer, in the watertable section of the action, also moves, tripping the "selected" ejector's release latch, so that when the gun's btoken open, any extractor/ejector so tripped will then be released to be driven backward fast/hard by it's own spring.
If one's not tripped, then it will only be cammed back enough to raise the unfired shell out of the chamber enough to be grasped for manual removal (about 3/8").
There could be a worn or broken part anywhere in that chain - but if both barrels fire, ergo they're geting cocked, so that end must be presumed to be ok, and I would look between the cocking rod for that side's hammer and the ejector/extractor.
With the barrel set off the frame, does the ejector move freely back in under manual pressure ?
If not, there may be a horrific bind in there, enough that the EJ spring can't overcome.