If my opponent plays Thousand Tailed Watcher and triggers its effect, can I respond to its ability after the -3 might effect resolves on the same chain?

No, you cannot respond to the Thousand-Tailed Watcher's "When you play me" (WYPM) ability after it has already resolved.

According to the RiftJudge FAQ (ID #5127 and #10506), once a triggered ability resolves, it is cleared from the chain and the game state moves forward. There is no opportunity to play reactions or actions after an ability has finished resolving unless a new item is placed on the chain or a new phase/game state is triggered.

Here is how the timing works for that scenario:

1. Playing the Unit: Your opponent plays Thousand-Tailed Watcher. It enters the battlefield. This does not use the chain and cannot be responded to.
2. Triggered Ability: Its WYPM ability is placed on the chain. The game enters a Closed State.
3. Reacting: At this point, the ability is on the chain. You (and your opponent) may play [Reaction] cards to interact with the game state.
4. Resolution: The chain resolves (LIFO). When the WYPM ability resolves, the -3 Might reduction is applied to all units currently on the board.
5. Chain Empty: Once the ability resolves, it is cleared from the chain. If no other items were on the chain, the state becomes Open.

Per the RiftJudge FAQ (#5127), "After the ability resolves, you don't have a chance to play an action." You must play your reactions while the ability is still pending on the chain. Once it resolves, the effect is finished and the window for responding to that specific item is closed.