No,
I was able to see that the account showed as Idle on another Buddy List (Like it should and like iChat versions did before for the previous 15 years).
The accounts do NOT log out but go Idle after the keyboard and mouse (or other input device) have not been used for fifteen minutes.
The unexpected bit was the app did make the Logging in sound when I activated the computer which activated the Screen.
As far as I could tell no account actually logged back in.
To Restate.
AIM for instance allow 5 states.
Available
Away
Invisible
Idle whilst Available and
Idle whilst Away
The top three of these can be set by the user.
The other two cannot.
The bottom one in fact is not used in iChat or Messages (you only see Away markers for others in your Buddy List and your app only sends Away flags)
This Hint tell you how to change the Idle time
http://hints.macworld.com/article.php?story=200402181751007
You have to change the ichatagent to imagent for iChat 6 and later.
I used a second computer with a separate AIM Account to see the accounts I have on my iMac so I could check they are logged in.
The App making the logging in sound is not indicative that the app has logged out (Unless the computer Sleeps).
7:39 pm Wednesday; May 27, 2015
iMac 2.5Ghz i5 2011 (Mavericks 10.9)
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