It's time to pay the bills......
Puppets in Toobworld are alive when they are seen interacting with humans independently of manipulators. The background splainin is that they are the Spirit People of the Isle of the Blessed* which has since become known as the Living Island. When a Spirit occupies a puppet shell to give itself shape, it assumes the characteristics of that form.
For all intents and purposes, Kermit is a frog, Kukla is a dragon, Chairy is a chair.
And the same holds true for puppets who look like established celebrities. They may know they are not the real thing, but still they continue the illusion by acting like the celebrity they resemble.
In this case, the puppet would always act as if it was the real Jimmy Durante.....
* The Isle of the Blessed can be found in the 2nd Century tome "True History" by Lucian of Samosota. The inhabitants of the island are invisible and so they wear spider webs dyed purple to give them shape.