@Sean
Both Endomondo and Strava are tracking your position only when you activate the exercise.
In Endomondo users get this feature for free - they can open their browsers or run the smartphone app to see which of their Endomondo friends do something and where they are. However, as you mentioned, Endomondo has no API available for public.
In case of Strava, when you start your activity you can select an option for Strava to send by text message to predefined contacts a link to a website with your activity, map and position online. It is called a Beacon and it is paid. Users have checked that in fact anyone who has the link can visit this page and see the location. Whereas all other features are well described in the API manual (http://strava.github.io/api/partner/) the Beacon feature is empty. Maybe if you are a premium member they will reveal the contents of this part.
In my opinion you are not limited to these two apps. There must be plenty of others, particularly those designed to track children etc.