The videos that you acess using any type of search inside the app is never collected nor is any personal information. All your personal preferences for search, filters, searches, watch history, etc. is never collected outside the app, nor is distributed to any other third party outside the app. Everything that you do is stay inside the app.
Search & Save app. requires the permission to use the "Internet Client" capability in order to provide access to the search results. The app is using the official YouTube API provided from Google to make the search or to provide you with the play features.