We let some peonies form seed pods after they have bloomed. Generally, we deadhead the flowers (cut off the spent flower head) at the end of the bloom season. If you don't deadhead, some varieties will form seed pods over the summer. Seed will develop in the pods (not all varieties/pods contain seed) and comes in a variety of colors. When the pods start to split open, we harvest the seed and sow it in the fall.
Some peonies, like 'Coral Charm' and 'Coral Sunset' produce beautiful pods; however, finding a seed on these varieties can be 'one in a million'. It has happened for other growers; although I can't recall finding one in 20 years. If I did, it was a fluke and likely that special one I slipped in a pocket, telling myself I would remember; only to be tossed in the washer, never to be seen again.
Other varieties produce seed in abundance, like 'Mischief', a single pink peony pictured below. Like many other single-style peonies (their flowers have only one row of petals), 'Mischief' produces seedlings that are single, double, semi-double and Japanese flower forms. It's always a treat to see what you get.
Here in the Willamette Valley of Oregon, we are blessed with flowering seedlings only 2 - 3 springs after sowing. Other areas of the country may take several years longer.
All the seedlings pictured here are from 'Mischief'. Notice the color and flower formation variances. These were open pollinated by bees.