I went back Friday afternoon to take some pictures and collect some of the fruit. I picked up two gallon-size ziploc bags of fallen persimmons, and tried a few of the ripe ones. They tasted pretty good, but not as good as a fully ripened Fuyu. It seems that the best tasting ones were just a few minutes from rotten.
When I got home, I extracted all the seeds. The seeds were covered with a slimy film or seed coat, which I assume is to protect the seed and keep it from drying out over the winter. I removed the seed coats and spread the seeds out on a paper towel to dry. After they dried out for a while, I wrapped them up in slightly damp paper towels, placed them in ziploc bags, and placed them in the refrigerator. According to the CRFG Fruit Facts, persimmon seeds should be cold stratified, so these will stay in the fridge until they're ready to plant in the spring.