We started growing peonies from wild collected, Josef Halda seed ~ 15 years ago. Regardless of the species, they can be handled in a similar fashion. Early on I would soak the seed in 35% hydrogen peroxide - a very strong bleaching agent that will soften the seed coat. Leaving the seed in for ~10-30 minutes is sufficient. Planted in a some standard seedling mix, the seeds will swell, forming a small radicle (root only) during the cold months of fall and winter. Germination will happen the following spring.
In later years, I have changed over to the standard I use for almost all the seed I receive - soaking in GA-3 until the seed swells. The only extra step is rubbing the coat of the peony seed on sandpaper to break through the hard, waxy cuticle. I'm not sure if the GA-3 is necessary, but it doesn't hurt.
Germinating seedlings should be left in their pots for at least one full growing season - meaning they can be transplanted into individual pots in the fall if so desired. Otherwise, leave them in the original pots, but fertilize with some slow-release pellets for another season. The larger the roots are, the better the transplant will be. Like peony divisions, you can handle them easily in the autumn, leaving them bare-root for hours , even days without harm. Like other ranunculacaea, peonies do not like pot-growing. A mix based on composted bark with added grit and sterilized loam works best - addition of loam really helps. They are "feeders" and will respond to fertilizer. Such as a general purpose 15-15-15, preferably in slow release form. Pot grown for ~2- 3 years, they will be big enough for garden planting. In the garden, heavier, loamy soils are definitely preferred. From pots, they can be planted almost anytime as conditions are favourable.
Collecting your own seed is lots of fun, but be aware that hybrids may occur. With the woody peonies this is less a concern as most are derivatives of p. suffruticosa. The herbaceous species are more of a problem, indeed in nature hybrids occur between different species growing in proximity. In the garden you may have to take isolation measures if you wish to have pure seed.
The seed pods themselves are very decorative, having a thick, corky capsule that splits open to reveal the shiny, black (or dark brown) seed. Brilliant red "seeds" are barren. Don't keep them. They do add a splash of colour.
Besides the pleasure one derives from growing plants from seed, there is the added knowledge that they will be virus free - this is a big problem in the industry. Josef Halda, whose numerous trips to China were funded by specialist growers in the Netherlands commented once that these growers told him that the seed grown stock was so vigorous that they went on a crash program to clean-up their old "named cultivar" stock. They kept only the best of the old stock, putting it through tissue culture. Inferior varieties they simply trashed. The new stock was that much better.