Do I Need an Internet Connection to Fix My Photos?
No. The work happens on your computer. One optional feature uses the network, and you can leave it off.
No. Reading a .json file and writing a date into a photo is work your own computer does easily. Nothing needs a server.
That matters more than it sounds. The alternative, uploading your whole photo library to a website, means handing every family picture you own to a stranger's storage. It is also impractical: a 300 GB export is not something you upload twice.
One feature does use the network, and only if you switch it on. Turning coordinates into a place name like Paris means asking a maps service. Only the rough coordinates are sent, never a photo and never a filename.
Leave place names off and the app makes no network requests at all. You can disconnect entirely and it works exactly the same. There is no account, no sign-in and nothing uploaded. The first 500 photos are free.