Ah, well its a bit more complicated than you'd think - some persimmons are astringent, and some aren't, and the harvesting/storing procedure is slightly different for both. Without knowing which one you've got (and you may not know yourself) then its probably going to be trial and error to some extent. Yours look to be the right size for Oriental persimmons, but they are also split between astringent and non astringent. If you have a label or the botanical name for your variety of Persimmon, that could help to decide which type you've got. Link below gives more information
Judging by the shape and size of the fruits, they may be the sweet, non astringent ones - another link below showing already harvested fruits similar in shape to yours, but there are hundreds of varieties...