Why are my messages returning to me?

On my iphone it looks like the person I message sends the same message back

That might be because the person is replying with the email string attached.

I think you always get a copy of messages you send when you use conversations, the stringing of messages into threads.

It can be disabled.

Mouse over the Settings menu icon Gear Icon | select Settings.
Click Viewing email.
Uncheck the box to deselect the Enable conversations option.
Click Save.