There are often more than one way to fetch data. Take finds for example.
Via the API Project-GC can ask for a specific user's finds. It's my understanding that such request will be denied if the relevant privacy setting is enabled. However, there are other api methods to fetch finds for a geocache, in that case it won't be denied. Project-GC uses both methods, where one should still work. But by using the privacy setting that users updates might be slower.
Regarding Gallery Project-GC only fetches that from Geocaches, not from the user's profile. This will probably be uneffected.
For souvenirs, the result of using the Privacy setting will be that the user has zero souvenirs on Project-GC. Affecting Profile stats and Challenge checkers.