For the healthiest thriving plants, most definitely you need CO2. A definite must. CO2 is the difference between 'having' plants and 'growing' plants.
Yes, big fat goldfish do give off a lot of CO2, but they use up a lot of oxygen which requires a lot of air bubble agitation, which in turn escapes the CO2 off through the surface before the plants can use it. So you need more CO2 being injected than is being lost by your air bubble agitation, (which is why the ideal planted tanks have few or no fish at all so don't need air agitation).
My small CO2 cylinder ran out of gas a few weeks ago and I'm trying to find time to go get a larger one (the gas rental place* is a long drive away which is why I haven't been able to find the time yet). Anyhoo... in the meantime, my plants have quickly gone from a rich, thriving, dark-green jungle to sickly, pale-green spindly plants struggling to even survive now whereas before I was pruning them back once a week they were growing so lushly.
With everything else being equal (ie plant food etc), my own experience over the past few weeks means I can safely say that CO2 makes a MASSIVE difference. Yes, MASSIVE in capital letters. The difference between with and without CO2 is like chalk and cheese and I myself will be getting the largest CO2 cylinder rental as possible to ensure I never ever run out again!
But you only really need CO2 if you want a serious heavily planted tank. If you only really want just one or two plants, it's a lot cheaper and easier to just replace them every now and then when they perish.

*I just have to go there initially myself to begin with to open an account and be sure my regulator fits onto their commercial bottles of CO2, but after that the replacement cylinders are home delivered just like pizza! (which will only be about once or twice a year though at most)