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The OP's definition of "work" went beyond whether the AirPods connected directly to the Apple Watch:

>As in, if I leave my phone at home, will I be able to stream music and calls from my watch to my AirPods while I go for a run?

For my response, I assumed two things:

(1)"Streaming music" meant listening to music that is not stored on the device.

(2) "Calls from my watch to AirPods" meant that during a run, OP was not in constant range of a remembered, open WiFi network.