Acceptable bits for western or engish are really only determined by show rules.
In western pleasure, a horse that is under 6 years old may wear an O-ring or D-ring snaffle for showing. Also, the riders may use two hands. When the horse turns six though, you have to use a shanked bit and ride single handed in show classes.
The reason for the two handed rule is that snaffles are basically useless without two hands, so riding one handed in a snaffle defeats the purpose.
For english, shanks are prohibited, so riders will only use english snaffles, kimberwicks, or pelhams. I don't think there are any age related rules.
So basically what I'm getting at is that you can ride in a snaffle, but if your horse is 6 years old or older you can't show western in one.