Photo Deduplicator is a simple utility: it takes one or more input files (image files) – which you provide –, applies the renaming rule that you define and creates copies of all input files in the output directory you have selected beforehand.
It fully complies with Apple's "sandboxing" requirements for Mac applications – otherwise it wouldn't be in the AppStore anyways.
In particular, Photo Deduplicator does ...
Instead, you control ...
For the sake of transparency, there has to be noted that Photo Deduplicator will check for newer versions of itself being available at the Apple App Store for download. In order to achieve this, Photo Deduplicator issues two HTTP requests whenever you open the application. The first request goes to the Apple App Store API to retrieve the version number of the latest available Photo Deduplicator version. The second one goes to the Photo Deduplicator support website (the site you are currently browsing) in order to load the release notes for all Photo Deduplicator versions. In combination, the responses to these two requests are used to determine whether you will be presented information about the new version being available and what's to be expected from it, plus a link to the App Store to review and potentially decide to update your Mac with the latest Photo Deduplicator release.
There are no other network requests issued and there is no information transferred from your Mac to either of these sites besides the implicit fact that apparently Photo Deduplicator was started and an HTTP request was sent from some IP address (the external IP address your Internet provider has currently assigned to your access point to their network). There is no (!) personal data transferred.
If you don't like the idea of Photo Deduplicator automatically triggering HTTP requests to the Apple App Store and the Photo Deduplicator support website, you can switch off this version check completely: Either by deselecting the corresponding option in the welcome screen of Photo Deduplicator when started for the first time after installation, or at any later point in time in Photo Deduplicator's settings (see Settings).
Here's the two static HTTP requests issued by Photo Deduplicator for your reference. Feel free to check those out for yourself – it's two JSON-formatted documents being linked.