Your assumption is correct: Z-tailwind only affects the Pokemon who used it.
To be specific, the critical-hit boosting effect of Z-moves are treated similarly to focus energy, and this applies to every Z-move where applicable. A Z-Move that boosts Attack by two stages as an additional effect is functionally giving you a free Swords Dance boost, for example. Both critical-hit ratio boosts and attack boosts are lost when you can switch out. You can still get the effect transferred by Baton Pass if you choose to, but that's about it really.