I left my phone in my car and went for a walk, had my Bluetooth off and it worked well. I'm not very tech savvy but it appeared to work fine.
It will record distance and pace using the accelerometer in your watch currently, but will not provide a map of your route. Runkeeper has worked without using your phone for a while, just doesn't use GPS. There are no apps other than the native Workout app that use the integrated GPS in the Series 2 currently.
Did you get a map of your route in the Runkeeper app?