I have been germinating persimmon seeds, with mixed results, for a year or two. I love persimmon!
From reading: Best is to put seed straight in the ground, while still wet. If it's summer, put the pot of dirt in the freezer for about 30 days to cold stratify. (if it's winter, just put it outside and wait).
From experience: Putting in the freezer after drying (preferably in damp soil) works to spur germination, but I still only get about 1 in 10 seeds to grow. When they grow, they're susceptible to rot, so I've killed some by overwatering... But I have a couple that are growing... slowly... because I'm so scared to overwater. It may well be much, much easier in more temperate climates.
I don't think they necessarily grow true to type-- I collect seeds from the fallen fruit around my mother-in-law's apartment complex in Korea each year when we visit, and I notice sometimes fuyu persimmons have clearly crossed with Hachiya or kaki (long versus round seeds).
sortof-almost-off-grid in South Africa: www.concretegardener.com
Fall before last I collected several seeds from under a wild persimmon tree in a field. I took them home and planted them in a little planter and almost all of them sprouted. I planted about 6 or 8 (I guess I'd better go see how many I planted) in the front yard after they were about 6 inches high. They just kind of survived the first summer after I planted them until this spring. They all survived though and are finally growing. I planted a couple of asian persimmons last fall hoping to graft onto my wild persimmons next spring. The asians are also just putt putting along at this time, so I may need to wait one more year to get enough asian to persimmon to do the job.