There are some horses that do like to be able to "carry" the bit themselves. I think it was Cowchick that posted a picture of a horse a while back that was holding a spade bit in their mouth that didn't even have a bridle on it.
In some circles, that is what people want horses to do because carrying the bit allows them to feel even the most subtle changes in the bit's balance and respond.
However, having a horse that will do that greatly depends on the individual horse and the training. I don't really know how other
folks train a horse to carry the bit, but I generally leave my bits a bit lower in the mouth than most people...even snaffles. That has seemed to do the trick with the horses that I ride and encourage them to pick the bit up and carry it where it's most comfortable for them.
Also, good hands are really important in getting a horse to carry the bit. If you abuse a horse's mouth, they won't carry the bit because they start trying to avoid all contact with it. So, they just let it hang there by the bridle and try to ignore it until it "bites" them. Hence, the creation of hard mouthed horses.
A good sign that your horse might be carrying the bit is that they hold onto it when you go to unbridle them. Verona posted a thread about this Holding on to the bit
Once I move a horse up to a curb bit, I'll often play with the bridle, taking it up and down holes until I find that sweet spot where the horse seems to be happiest. Dobe, I have to leave his bridle hanging down a little over 1/4 inch from the corner of his mouth or he gets huffy about it LOL