Forgive my double post, but I have made some appreciable progress in fixing this bug across four devices.
I am of the opinion that this is a webOS Account/Palm Profile bug. I am not certain how it manifests itself. I originally tried to make a new Palm Profile after experiencing this bug back in December of 2011 on my Pre 2, but that profile too had the bug. Someone here on webOSnation suggested it was a hardware issue, so I always assumed that was the case.
I recently purchased two HP Veers and an HP Pre 3. All three devices had the bug when I attempted to use Skype (Veers were updated to 2.2.4). The chance of having four devices with this issue, if it was hardware related, seemed highly unlikely to me. I created yet another new Palm Profile on one of the Veers and to my surprise the bug was gone! No double messages using my Skype account! A hardware issue seemed less likely...
I signed into the new Profile on my second Veer; no issue with Skype. Tried with Pre 2; nothing wrong. Tried with the Pre 3 and now I had the bug again. I created yet another Palm Profile on the Pre 3 but it also had the Skype bug. I tried to sign back in with both of these newer accounts on the other devices and both had the bug now on all devices. I was certain it was a Palm Profile issue.
Steps I Used to Fix The Bug (worked on all of my 2.2.4 devices and all my various Palm Profiles):
1*) Turn off and erase Palm Profile Backup
Preferences > Backup > Slider to OFF > Select Erase option
*Sometimes after the erase procedure completed, the Backup slider would reset to ON. Keep turning it to OFF and selecting the Erase option until slider stays in the OFF position.
2) Fully Erase and Reset Device
Preferences > Device Info > Reset Options > Full Erase
3) Setup device with same Palm Profile from Step 1
4*) Sign in to Skype and send a test IM
*Some of my devices were fixed after this step. If your device is normal now, stop. If issue persists, continue below.
5) Repeat Step 1 (Make sure Backup is off and select the Erase option)
6) Erase Apps & Data + Reset Device
Preferences > Device Info > Reset Options > Erase Apps & Data
7) Sign in to Palm Profile after Reset
8) Sign into Skype and send test IM
All of my devices and profiles were fixed either after step 5 or 8. Please note that you will lose any saved data that is only associated with your Palm Profile if you turn off and erase the backup and then reset your device (ie. text message history). App Catalog purchases can be redownloaded after device setup completes.
This worked for me across multiple devices and Palm Profiles. Hope this helps!