It's probably in a jar at his hideout like all of the Sharingan eyes that he has.
There are significant benefits to being able to use both eyes simultaneously. The Sharingan clearly beats out the Rinnegan in terms of unique abilities, whereas the Rinnegan allows access to apparently every ninjutsu (except Kekkei Genkai). If Obito had removed his other Sharingan, he would have probably lost his ability to use Kamui, which means that he wouldn't have access to the other dimension, he wouldn't be able to become intangible, and he wouldn't be able to travel quickly between multiple locations (unless that is an ability that he gets from Zetsu).