I'd like to check if I understood well. Both expressions have the same meaning, and can be used for both people and things, with the difference that "been missing" must be followed by a period of time which indicates the length of absence and "gone missing" can never be followed by a period of time.
Have you seen my shoes?
-They have gone missing.
-[STRIKE]They have been missing.[/STRIKE]
-They have been missing for weeks.
I do not know where my dog is.
-It has gone missing.
[STRIKE]-It's been missing.[/STRIKE]
-It has been missing since Friday.