I think while a ship's computer is in range then it's computing power can be used to amplify computing power of various devices, although I don't think that is something that is ever specifically stated, but there's been plenty of times where things like tricorders have been used without a ship nearby. As for communicators, there must have been a time when crewmembers used them to communicate with each other without a ship nearby but I can't think of a specific episode at the moment.
My point was that I don't think the commbadge is really much of a computing device on its own. I think It's more of a relay device. At least that's been my interpretation of things based on their behavior. Tricorders on the other hand would definitely have onboard computing, especially when you consider the size of the original tricorders from the original series and their onboard database. And if there are instances of commbadges working without a ship nearby, it's possible they use a tricorder's power similar to a wifi hotspot.
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