Treat them like any other seed requiring a cold stratification, but here are some pointers:
I let the fruit get really ripe and soft, then separate the seed from the flesh in a bucket of water.
In Oregon I just plant them 1/4" deep outside in pumice in December, but if it is extremely cold where you are you may keep them in a cold frame. I believe the climate is mild in their native range.
Princess persimmon germinate very late: Late June or even July or August. Do not panic this is normal.
It may take up to 5-7 years for the seedling to flower. They say the ratio is 60% male and 40% female.
This is January:
View attachment 462385
... and June:
View attachment 462384