I have seen this question and there generally very little information on this topic. In the question linked above the they talk about the activity app. I use nike+ running. Last time I went for a run I ran for some time with my phone in my pocket, then I ran without it and the watch would not pick up the distance or any other info during the run. I started the run from my watch so I thought it should not be problem. Has anyone figure this one out yet?
I think Apple's included Workout app can record activity and heart-rate data without the iPhone, but the Nike+ Running app can't (as of the 17th of November 2015).
Here's the info page from the iPhone Nike+ Running app (Nike+ Running > Settings > Apple Watch)..
Nike state that you need to have your iPhone on you.