Can you tell me if you have noticed any dragging of the rear end on the dog that is being licked?
Have you noticed any unusual strong odor from the dog being licked?
Are the dogs neutered or spayed?
How long has this been happening?
I am so sorry that your dogs are worrying you. There are two possibilities that come immediately to mind. The first would be the anal glands on the dog that is being licked need to be expressed. This is a common problem and your puppy may be sensing them, as they are scent glands used for marking territory. Your Vet can express the glands and that should slow or stop your puppy from this behavior, if it is anal glands.
The second issue is your dog may have a urinary tract infection brewing and the puppy may be sensing this and trying to help. Dogs have a very strong ability to smell things you or I would not even notice. That is why dogs are being trained for all types of scent work to include bomb sniffing, finding lost children and even sniffing for Cancer. I do suggest a Vet visit to rule out the anal gland issue and Urinary tract infection. Joan