There used to be a general rule of leaving one finger's width of space between your horse's lips and the cheeks of the bit. When all you could get was single link bits, this rule stood pretty firm, but now there are a variety of different mouthpieces and the fit will vary slightly with each different type. I'm going to go through a few options here so that you can be sure that you new bit is a good fit.
Lozenge/ french link/ double link bits
The reason for the one finger rule is that the single join bit folds in mouth link a hinge, and so the cheeks come closer in together. Allowing that bit of room helps prevent any pinching of rubbing from this folding action.
Lozenge or french link style mouthpieces don't have this folding action. Instead, when you take a contact they simply depress the tongue and the cheeks move away from the lips, not closer in. So, it's important that you don't have too much room to start with when the bit is at rest, or with a contact you will end up with too much bit. Too big a bit and you have excessive side to side movement in the mouth, as well as generally being sloppy and uncomfortable in the mouth.
So, as long as the lips are sitting clear of the bevel holes in a loose ring, then with a contact there will be plenty of room.
Left: Lozenge style bit too small, Right: Just big enough- the lips are clear of the bevel hole for the ring, and with an even contact from the rider will only move further away from the lips.
Mullen Mouth or Ported Bits
Any mullen mouth (or ported) bit- anything with no join at all- need to sit fairly flush with the sides of the face. Most horses will take a slightly smaller size in a mullen bit than their jointed bit, as there is no need to allow for the bend in the mouth as the jointed bit folds around the jaw. Too big, and the mullen or ported bit will again slide from side to side in the mouth, which is distraction and with a port, can be very uncomfortable.
These are a slightly different kettle of fish. They are designed specifically to protect the sides of the lips and cheeks and so are shaped to fit showing more mouthpiece at the sides than you would expect. A Myler should not sit flush with the cheeks, but nor should you be able to see the "elbow" bend in the mouthpiece either. Most Myler styles are not available in a wide range of sizes, so it may be harder to get the right fit for your horse in one of these.
Left: You can see the "elbow" of this Myler bit showing, so it's too big. Right: a well fitting Myler- there will be more mouthpiece showing at the sides than you might expect, even with a fixed cheek such as this.
Single Join bits
Yes, you still need a finger's width on both sides, between the lips and the cheeks of the bit, being a loose ring or fixed cheek bit.
Loose ring bits compared with Eggbutts
A correctly fitting loose ring, evenly supported by the rider's contact, should not offer any more risk of pinching than an eggbutt cheek. However, some horses are more fleshy in the lips than others, and some riders consistently heavier in one hand than the other (pulling the bit through slightly to one side) and in these cases you may be better in a fixed cheek bit for protection.
I am not a fan of Bit Guards in this instance, as they generally just make the bit fit "smaller" by taking up more room, and the rubber can increase friction and rubbing rather than decrease it.
A nice fitting NS lozenge style eggbutt snaffle.
Double Bit sets
As a general rule of thumb, your bridoon size is the same as your normal snaffle bit size, and your weymouth bit will be a quarter inch size smaller. Your weymouth is a ported (unjointed) bit, and so needs to sit quite flush, and it also sits on the slightly narrower, lower part of the jaw. Your bridoon will generally sit in the same place as your snaffle, though some horses fit the double better with their bridoon a quarter inch bigger to give a little more room for the two bits in a smaller mouth or those with a shorter lip line.
These are general rules and guidelines, but remember that every horse is different. Sometimes you just need to try things on to see if they will fit- just like clothes! Here at Bit Bank Australia, as long as you try your bits on promptly after receiving them to check, if they are the wrong size they can be exchanged for the right size. Just be sure to return them clean, and with the tags and packaging.