How to open a GUI for a the tamer of a custom mob and only the tamer?

It's hard to explain, so let me explain it like this:

You have a mob, that when right-clicked, does nothing

You shove a button down it's throat to tame it (lol)

You right-click on it again, and it opens an inventory

You add an item and close the inventory

You open it again and you can take the item back out

Someone else right-clicks on it, it doesn't open for them

How would I do that? I have done the tamer only bit but I can't store the items in the GUI...