Mine live and propagate in just the water, but they are not upright and beautiful as they are when planted. If they aren't being knocked around by current, they will typically root into sludge or dirt on the bottom.
There are many different plants called "Elephant Ears". It would be good to know which ones you are specifically talking about as it will allow people to provide the correct advice for the correct plant. Most Colocasia and some Alocasia will grow in water provided it's not too deep. Others prefer shallow water, or even just very wet soil. And some Alocasia will only tolerate well draining soil. Other Elephant ears such as Cyrtosperma and Typhonodorum will sit happily in a lot of water, but Xanthosoma must be well drained. None of them like totally dry soil.
Don't remember what variety mine are but I just stick them in a pot with rocks and lower them into the water to top of pot.
They do great but sometime get so big they flop over and I then cut them off cause fish nibble on them and they look ragged.
All around Florida, elephant ears (and water lilys) grow in the retention ditches, in mucky water. The counties have a big machine that goes around and routs them out. If you want some of those, I'm sure one of us could box some up and send them to you for postage. They ARE INVASIVE.
My Black Magics also did well just loose. Got huge, in fact, and were really pretty. I did find, though, that I had more winter kill that year with the roots exposed. Probably best to pot them up at least for the cold months.
I also have a water pot, separate from my pond, where I grow several varieties. It was originally a lotus pot that ended up full of EE pups instead, so it's mostly compost with about 4 inches of water. No filtration or anything. They all do well there too. Truth is, I have much better luck growing them in water than I do in the ground.
Well, I find I have Colocasia Esculenta black magic. Thanks to yout comments, I am going to try it in a big caldron - 36" across, about 2 feet deep. I,ll keep its
It just under the water level. If I have success I'll post a picture.