I do both. Mine will shine just as much if I use a blow dryer or not. The difference is how the hair lies. If I blow dry, it will be nice and perfectly straight. If I let it dry naturally, it will wind up with a slight wave and will not lie as nicely. I let them dry naturally quite often. But when I want them all pretty, I make sure to blow dry and keep brushing through as I do, just as I would my own. In fact... I use a little bit of my own Redken Smooth Down protectorant on them when I blow dry so as not to have them get hair that is burned from too much hot air. I have never used my flat iron on them -- I don't need to, to get them straight; but I have heard of some that do.
It is all in what you are wanting them to look like when done and how much you want to let them just be natural. I only have one of mine in long hair right now. Annie is short and I may keep her like that. I sort of like how she looks like a puppy again and she is much easier to maintain like this. She has a fine hair that required a lot of work when long. Ben is easy, as long as I don't let him get to the floor. Then he brings in all sorts of grass and debris from outside. So, I try to keep him at least an inch off the ground. But he just doesn't tangle or anything. Combs out like a breeze and a blow dry leaves him looking like a prince!
I am so partial!