Hi,
It unfortunately the way things can be for some people.
However if it works in one Mac User we know then that the app works, that the Serial Number is not the issue, and that it is something about the regular Mac User Account that you use.
There are several potential .plists (preferences files) it could be as well as a faulty keychain item or even a turned Off Font (or one with an issue).
Lets set up some basic but normally unused maintenance things first to give use the best chance.
Some of these you may have tried already.
In Applications/Utilities launch the Disk Utility app.
Highlight the boot Drive on the left hand list and make sure the tabs across on the right are set to First Aid.
Now click the Repair Permission button (lower left of the right hand pane if that make sense).
This consults a data base on your Mac that is updated when you install or Update apps and the OS (and therefore more up-to-date than running Disk Utility in Recovery Mode)
Test the Messages app.
Once this has finished Shut Down the computer.
Restart it holding down the CMD+ALT+P+R keys (PRAM/NVRAM rest) Until you have heard three Start up Bongs - then let the Start up continue.
Test the Messages app
Follow that with a Safe Mode Start
You can normally access your Own Mac account (some things are slower and not used so some functions are restricted)
Try Messages again.
These last two do clear out some caches that cannot be cleared in other ways.
Next open the Font Book and check these three Fonts (again in some cases)
Helvetica, Lucida Grande and AppleGothic (all one word)
None should be duplicated. All should obviously Validate (File Menu). Keep a look out for Helvetica Fractions as this has caused issues and conflicts in the past.
If anything needs sorting then test Messages afterwards.
If you still have issue with the Messages app logging in the iMessages account navigate to ~/Library/Preferences
Find:-
com.apple.ids.service.com.apple.madrid.plist
com.apple.ids.service.com.apple.iMessage.plist and
com.apple.imessage.bag.plist and Drag them to the Trash.
These three are involved in the IDs you log in with and the iMessages server your app connects with.
Restart the Messages app.
Report back at this point.
We can still do a Keychain repair, delete the IDS item from the Keychain to force it to make a new one and some Sandboxed .plists in ~/Library/Containers that could be looked at.
7:45 PM Saturday; November 22, 2014
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