Natural bodies of water with plants, algae, and circulation are capable of handling a certain volume of organic waste from fish and wildlife via biological filtration. Artificial ponds, in order to support life in them, need to mimic natural features.
Your ponds may be on the threshold of the size needed for your fish volume, but there might not be much elbow room in them. Plants will provide natural filtration, but circulation is also a good thing to have to avoid stagnant water and a good rule of thumb there is that you should have a pump that is capable of circulating the entire water supply every 1 to 2 hours. So, if you have a 1000 gallons in the pond, a 500 gph pump is the minimum you want.
Still, in addition to plant filtration, you should filter the water. I think you could probably combine the pump and filter for smaller artificial ponds and you want to have a biological system to encourage the nitrogen cycle. I suspect your ponds are more like big fish tanks, so you could approach it from that angle.