Basically this is the way I see it. I can't be bothered to do the science because I don't particularly like the effect of an airstone and all my tanks are planted, but please do tell me what you think about this reasoning.
An airstone increases the exchange of gas between the air and the water. That means that the oxygen depleted by the fish while breathing will be replenished more quickly.
That also means that the CO2 depleted by the plants would be replaced more quickly, if the fish didn't contribute CO2 through breathing. How this balances out, depends very much on the balance of light, nutrients, plants and fish in each aquarium.
If you are adding CO2 to the water you could
- Strike the balance and therefore the extra gas exchange would make no difference. You are adding as much CO2 as the plants are depleting minus the CO2 the fish are contributing.
- Add less CO2, which would mean that the airstone would actually help the water absorb CO2 from the air. That also means that the CO2 you are adding is wasted, as the gas exchange with the air would take care of that
- You are adding more CO2 than the simple gas exchange would allow, therefore having more CO2 in the water that you would have with the simple gas exchange with the air. If you increase the gas exchange then this extra CO2 will be dissipated.
In fairness cases 1 and 2 are pointless, and by that I mean that whether you have CO2 injection or not that will make no difference to the CO2 content in the water, so they should not be considered.
Case 3 is the only case where injecting CO2 makes sense, as you would have a result. That also means that if you
are increasing the gas exchange with the air, the water will be releasing more of the extra CO2 that you have been adding.