Ethan wears a collar and tags 24/7, when we adopted it said in the contract-thing that it was a requirement. On his collar he has his tags. One is his rabies vaccinations, and it has the number to our vet on their, so our vet could easily recognize him if someone brought Ethan to him (recognize him as the skittish dog that likes to poo' in his waiting room^^). Also, we have a tag that his shelter gave us. It was a number to their shelter and a certain i.d. number that the shelter can use to find out exactly who owns him.
Frisk doesn't wear his harness (his head is too little, collars come right off ) unless we're on a walk or in a public place, and all he has on his is his rabies vaccination, same as Ethan's.
We plan on getting a tag for both of them that has their name and our address and/or phone number.