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Neither of which requires running a server.
It all depends on how you implement it. He describes the text from images as using machine learning, which implies using an external API to do that translation.
Storing cloud backups requires a server and storage, etc.
It doesn’t. You can have local models.
Yes, it doesn’t have to be the author’s storage. Many, many apps implement it as adding the data to your Google Drive / Dropbox / whatever.
You can have local models, but the vast majority of people are going to farm that out to a 3rd party provider via API calls because it’s the easiest thing to do.
Remember I said “It all depends on how you implement it”.