If I take my iPhone on a run with an Apple Watch without GPS the workout app on the watch will use my iPhone's GPS data. Is that same true for barometric data now that the workout app uses the barometer on Apple Watch Series 3 devices?
Yes, it already works with watchOS 3 and iPhones with built-in barometers (iPhone 6, 6 Plus, 6s, 6s Plus, 7, 7 Plus, 8, 8 Plus, X) running iOS 10.x or newer.
Start an outdoor walk workout and walk uphill. Once you stop the workout you will see the elevation gain on the Apple Watch and in your iPhone's Activity.app.