The description of the Ersatz Eye from Xanathar's Guide to Everything reads:
This artificial eye replaces a real one that was lost or removed. While the ersatz eye is embedded in your eye socket, it can’t be removed by anyone other than you, and you can see through the tiny orb as though it were a normal eye.
Upon reading this, my assumption was that a user attuned to it can see through it at all times. However, the sentence is written in what seems to be the most ambiguous way possible. If we take "and" as joining two independent clauses, then the sentence is equivalent to:
"You can see through the tiny orb as though it were a normal eye, and while the ersatz eye is embedded in your eye socket, it can’t be removed by anyone other than you."
This would imply that the user can see through it whether or not it's in their head. There are other ways to restructure this sentence with that implication - but there are certainly ways to read it which imply the opposite. If there was no comma before "and", or if there was a semicolon instead of a comma, then it would be a different matter. However, I believe that rules of English grammar alone will not suffice to resolve this.
Is there any reason to conclude that the eye cannot be seen through while it is removed from the user's head?
It seems to me that this ability is commensurate with an attunement slot in 5e. Its power would be handily limited by the requirements for attunement - if it's left somewhere for spying purposes, the user would have to be within 100 feet of it at least once every 24 hours. I can't say if it fits the power of a Common magic item, though. There are very few Common items which require attunement in 5e, though, and the only directly comparable ones seem to be from Eberron.