Most of the things happening on Android Wear smartwatches are entirely dependent on the connected phone, but there is support for the watches to be a little more independent. The Sony SmartWatch 3, for example, has its own GPS chip that was enabled by the Android 4.4W.2 update. Now the popular fitness app RunKeeper has been updated to take advantage of it.
With the new version of the app, you can start RunKeeper on the SmartWatch 3 (as well as any future watches with GPS) and leave the phone at home. You won't have a data connection while away from the phone, but all your GPS data will sync to the phone app once you're back in range. As an aside, Android Wear also supports Bluetooth music streaming, so you can still have tunes while running without your phone.
This update also added support for SMS Audio BioSport Earbuds, which can measure your heart rate during your workout. That might be a good companion to the SW3, seeing as that device lacks a heart rate sensor.