I think that this has to go into bugs section, but I got a response:
"Sorry, but this is how our software works at the moment. I would recommend that you create a feature request for this" Here how chat request works now:
Client requests a chat support through web LiveSupport side:
/visitor/index.php?_m=livesupport&_a=startclientchat
- When client gets to a chat window that says "an operator will be with you shortly"
- and sends typed message before "operator has joined the chat conference"
- client gets alert pop-up saying - "Unable to deliver the chat message. Please wait for the chat session to begin."
- After this - the message that was entered by client disapears from a chat window
- so as you can see we have highly confused, amazed and what is really bad an angry client,because he would have to retype the same request from the beginning 
I'm asking everyone who want to keep their clients with a good support system to sign under this post, because I think this looks more like a bug that could make your clients have negative feelings about your support.