It happens with both, but the behavior is inconsistent and seems situational. Sometimes I will listening to local files (files in SkyDrive that are "available offline") and playback will be fine. Other times the same local files will not advance between songs once the screen goes black OR the songs will only advance after a long delay (30secs+ of no playback).
But the behavior is much more common when streaming. I will reiterate though, if I time out of a WiFi connection that requires some sort of verification to reestablish the connection, all functionality of the Xbox music ap ceases, even if I reconnect to the network. The ap will stop searching, won't be able to play even local songs, and just looses all functionality.
Hope this helps.