No -- there is no transaction that is involved in this messaging.
So, it's a messenger more or less. Good to know
Let's say that you bought a widget from me, and you have already sent me the bitcoin to pay for it, but you haven't given me your mailing address so that I can send you your widget. Let's also say that we don't know each other and the only information I have about you is the bitcoin address from which you sent payment, and an email address that I think is suspicous -- it may or may not be you on the other end of the email address.
So, I ask you to send me a "signed message" that contains your mailing address.
You then go into Bitcoin-Qt, under "Sign message", and enter the bitcoin address that was used in the purchase of the widget, then enter your mailing address in the text of the message, then generate the signature. You then email me those three things -- the bitcoin address, the text of the message, and the signature.
So, I ask you to send me a "signed message" that contains your mailing address.
You then go into Bitcoin-Qt, under "Sign message", and enter the bitcoin address that was used in the purchase of the widget, then enter your mailing address in the text of the message, then generate the signature. You then email me those three things -- the bitcoin address, the text of the message, and the signature.
Okay, I have a few visible and a few invisible addresses in my Bitcoin-Qt. Bitcoin-Qt interface shows me only the balance (which is not broken down by respective addresses of mine). I bought a widget from you and my BTCs were either sent to you from one address or from many addresses residing in my Bitcoin-Qt.
1. When I ''sign a message'' containing my e-mail address and send it to your public address, how does Bitcoin-Qt know which of my visible and invisible public address(es) to include in the message for you? - I do not know the address from which I sent you the money (when I click in Bitcoin-Qt ''Transactions'' then ''Transaction details'' it doesn't tell me which of my public addresses was used for making the payment in your favour) so I cannot enter my public address by myself (am I correct or not?)
2. Are you the only person who can see my message?