Skype does not work for a given account
Here is the solution provided by Microsoft:

Many issues are due to current MacOS permissions or other faults that are negatively affecting Skype and other programs (lag/improper execution); not necessarily Skype itself. Here are some possible solutions that can resolve this type of issue and other issues. Many people are successful with #2.


Create a new macOS user, log into that account, try to use Skype. If this works, this proves an issue that exists with the other account, possibly permissions or a startup program. - Since you have done this, so I strongly suggest you try step 2. 
Attempt to repair any permission problem in the failing user's account home folder.

Resolve issues caused by changing the permissions of items in your home folder 

Once repaired, try to use the latest version of Skype.
Attempt to remove every trace of Skype so that the latest version could be reinstalled without permission or other issues. This would include trashing and emptying any Skype named files located in the Application folder, ~/Library/Preferences, and what Finder could find. Be sure to retain a copy of any history files (main.db file and its folder), if you want to ensure the exportation of older history. - I don't recommend this
Manually Install Standard version of Skype for Mac.