What's the difference between ⋯ private ⋯ and ⋯ your eyes only ⋯?

If the sub's emote privacy is set to “private”, then emotes would look like this:

Domino inserts a key into Jane's LULU Cuffs, but it won't turn.                            ⋯ ｐｒｉｖａｔｅ ⋯

The “private” label means that this emote is only visible to the sub and any Owners who are nearby. Outsiders cannot see it.

In addition, some messages have the “your eyes only” label:

⋯ (Perhaps it's the wrong key?) ⋯                            ⋯ ｙｏｕｒ  ｅｙｅｓ  ｏｎｌｙ ⋯

This means that only you can see this message, and no one else.