Ah, someone who cares about privacy, nice! I think I have only good news for you:
First off, understand that with this app, you make actual and direct contributions to the OpenStreetMap.
Anything you contribute in this app is directly added to the map, there is no third party in-between the app and OSM infrastructure.
Editing OpenStreetMap anonymously is not possible. Any changes you make (and their date and location) are attributed to your OSM user account and publicly visible on the openstreetmap.org website. Because StreetComplete should only be used for on-site survey, this reveals where and when you have used the app.
The app does not share your GPS location with anyone. It is used to automatically scan for quests around your area and to focus (and thus download) the map around your location.
As mentioned, the app directly communicates with OSM infrastructure.
However, before uploading your changes, the app checks with a simple text file on my server whether it has been banned from uploading any changes. This is a precautionary measure to be able to keep versions of the app that turn out to have critical bugs from possibly corrupting OSM data.
To display the map, vector tiles are retrieved from JawgMaps. See their privacy statement for more information.
The data shown in your profile is aggregated from your publicly available contribution history to OpenStreetMap and then hosted on my server.
For some quests, some additional data from third parties is downloaded. In particular, the oneway quest queries an API, also hosted on westnordost.de, about likely candidates for oneways.
Photos you attach to a note are uploaded to my server and deleted some days after that note has been resolved. Their meta-data is stripped before upload.