Yes, the frogs do shed, as do the soles.
However, if the frog is really half missing, the horse could have thrush. Thrush can eat the frog completely away if left unattended.
If it is thrush, the frog will have a really bad odor to it when you put your nose down there.
Take a soft-bristled brush, and wash the frog area with betadyne and water. Pat dry.
You will need to apply thrush medicine to the frog(s). I use Thrushbuster. It is very expensive but it also takes very little. It has purple gentian in it --- same thing as is used for humans with mouth thrush.
If this horse has thrush like I'm thinking it does, believe me, if left unattended, the horse will go very lame, be in a lot of pain, and won't be able to be ridden until the frog(s) grow completely back.
While thrush is usually associated with mud, wet conditions, wet stalls, some horses can get thrush under any circumstances. I have one. I've fought chronic thrush with him the entire 11 years I've owned him, no matter where we lived, including the low desert of Southern California where there's hardly ever any rain
If you're hesitant about the thrush theory, the horse is sore and lame, half his frog is missing, a vet needs to see him ASAP and give his owners instructions on how to recognize hoof diseases before they become chronic
Please keep us posted as to what you find out is wrong with the horse