Wow! Two threads in and I find persimmons! I may be a tad partial...
While I love all persimmons, it really depends on what you want to do with them. Fuyu-type Asian persimmons (Diospyros kaki) (you may have some growing "naturalized near you depending on where you live) are great because you can eat them without waiting for them to be bletted (sort of like pears). The other great thing is that they can be peeled, cut into slices or chunks and used in ways softer persimmons can not be used. Hachiya-type D. kaki need to be bletted just like native (to the US anyway) Diospyros virginiana. The advantage is that they can be shipped everywhere and have a much longer harvest than D. virginiana.
But for FLAVOR, nothing beats D. virginiana. Even within the species, flavors run from pure sweet to caramel-like. The issue here is that they do not ship well (which keeps industrial ag away) and they are limited to eating fresh or in things which require pulp...like persimmon pudding. That said, you really can't beat them. They have a wide array of applications and for those of you who hunt or enjoy wildlife, you can't get a better wildlife tree...EVERYTHING that walks, crawls or flies, eats D. virginiana.
Someone above had it right when they said you judge ripeness by when the fruit looks like a small bag of jelly. This is correct. The skin will also almost let the pulp slip right out...sort of like the strength of a wet tissue. In fact, many will pop when they hit the ground.
Even if you don't judge them right...don't despair. I've been eating them all of my life...probably more than my weight each season, and I still get impatient or want to sample a tree I may not have the opportunity to return to. Unripe fruit can still be quite sweet, but the astringency which follows may be unpleasant. It is temporary though, so I don't sweat it.
Oh...and there is an unfortunate myth about having to wait until frost for ripening. It's a myth that has been countered by a good many folks for a couple of hundred years. People grow up hearing it and swear by it. Granted, D. virginiana ripens late in the year. Some trees can ripen VERY late. This doesn't mean they require frost, but they may get it. Many ripen much earlier. If you choose not to take advantage of the early ripeners, that's OK. Those of us who do will appreciate your convictions!