accelerometerapple-watchhealth.appiosiphone
We see that the Apple Watch sends step data from its accelerometers to the iPhone 6 (and above) which also has accelerometers.
When you walk holding both an (recent) Apple phone and Apple Watch – you trigger accelerometers on both. This raises the risk of duplicate data.
We can see data from both devices in the health App. 
My question is: What stops step data from the accelerometers on the phone and watch from doubling up?
The health data is automatically added by the watch. If you can see heart rate data, this means your Apple Watch is sending data to the Health app properly.
The message on the screenshot you posted is a bit misleading. In this answer you'll learn how to verify that the heart rate data is being transmitted correctly and how to shut off heart rate data collection.
Open the Health app on the iPhone. Select the Health Data tab. Navigate to Vitals > Heart Rate:
If you see data, that means your watch is properly sending data to the Health app. You can make this visible on the initial screen for the Health app by toggling the Show on Dashboard switch.
You can turn off Apple Watch health data collection. Open the Apple Watch app for iPhone. Select the My Watch tab. Navigate to Privacy > Motion & Fitness. There, you can turn off Heart Rate or Fitness Tracking:
As of watchOS 3, iOS 10, and macOS 10.12 Sierra, this is not possible, and the simultaneous rings are rather comical, if not downright annoying. What I did was:
My reasoning is that the only times I'm going to pick up a call on any device other than my iPhone is when I'm already using my iMac or iPad, but not carrying my iPhone with me. In those cases, the onscreen notification is sufficient. It's very rare that I'm near my iMac/iPad and not using either of them, yet too far from my iPhone to hear it ring. Those rare cases do not seem to fall under your use case.
This is not a perfect solution, but at least it will keep you from being bothered by ringtones that keep going off while you're already answering the call. The WATCH might keep tapping you for a second, but that isn't nearly as distracting when you're trying to talk to someone.
Best Answer
Apple Health is smart. It automatically aggregates step data, so that every minute, it takes the step count from the last 60 seconds from the device with the highest input and adds that to the daily total. It knows when the measurements from each device were started and when they ended, and it makes sure to not include any steps from two sources measured at the same time. This is a post-processing thing, not in the time of measurement: it's not that the Apple Watch shuts off the iPhone's accelerometer; rather, they both generate step data, as in your screenshot, but Health figures out which step counts to use from both in an aggregation-type process after the fact, getting you the most accurate and complete step count possible, while not exaggerating.