At this point, all DLC in the game is streaming (and thus requires online). They may add in a way to save songs locally but I don't think that has been announced yet if such a feature is coming at all.
When you buy DLC, you don't buy the song, but instead you buy the right to stream the song. If you don't have online then you can't stream, and if for any reason the song is removed from the game then you won't be able to stream it anymore either. For example, the A7X pre-order bonus will only be available for a few weeks and then it will be removed from the game and players will no longer be able to stream it. How many other songs have similar expiration dates is still unknown, but this type of thing might only relate to concerts and such (like the A7X DLC).