Hi,
The Messages app has Alerts in the Messages Menu > Preferences > Alerts.
The top item is a drop down of Actions.
The next line starts with a Play Sound box to be ticked and then a drop down of Sounds
There is then the Box for causing the icon in the Dock to bounce.
For the Invite variations Dialogue boxes do appear on the desktop in different places depending on the type of chat.
Test Chat ones tend to be top right of the screen.
Video chats are top centre.
Audio Only chats are to the right but below the Text invite spot.
This is where the Incoming ones appear.
You obviously don't get the invite box with out going stuff but the Chat is often where you left the last outgoing chat.
So you can set a sound to a particular actions and in some cases those could be the incoming invites.
They will work with iMessages and other IMs on AIM or Jabber accounts that you may have.
The Notifications is part of the System Preferences panes and has the options to have different types of notifications show up for some of the actions.
The items can be "Alerts" or "Banners" ( or None).
The so called Notification Center is top and far right in the Menu Bar where Spotlight used to be.
This can reprise old notifications for specific apps.
8:03 PM Tuesday; October 23, 2012
Please, if posting Logs, do not post any Log info after the line "Binary Images for iChat"
iMac 2.5Ghz 5i 2011 (Mountain Lion 10.8.2)
G4/1GhzDual MDD (Leopard 10.5.8)
MacBookPro 2Gb (Snow Leopard 10.6.8)
Mac OS X (10.6.8),
Couple of iPhones and an iPad
"Limit the Logs to the Bits above Binary Images." No, Seriously