What happens when I sacrifice a creature when my Teysa Karlov is on the battlefield?
Resolute Watchdog's ability is an activated ability not a triggered one, so it isn't affected by Teysa at all.

602.1. Activated abilities have a cost and an effect. They are written as “[Cost]: [Effect.] [Activation instructions (if any).]”
603.1. Triggered abilities have a trigger condition and an effect. They are written as “[When/Whenever/At] [trigger condition or event],
[effect]. [Instructions (if any).]”

As an addendum, sacrificing a creature (typically) counts as it dying, so your friend is wrong.  So sacrificing a creature could potentially cause Teysa's ability do something.

700.4. The term dies means “is put into a graveyard from the battlefield.”