If I play Bone Skewer, my opponent reveals their hand and I choose their Thousand Tail Watcher, playing it to a battlefield that I control, does the ability of the Thousand Tail Watcher give -3 to all my units, or does the ability not do its effect because I'm playing it?

When you play Thousand-Tailed Watcher via the effect of Bone Skewer, its "When you play me" triggered ability will activate, and it will reduce the Might of your units.

According to the RiftJudge FAQ (ID #10720), here is the breakdown of why this interaction functions this way:

1. Controller vs. Owner: Even though you are using Bone Skewer to force the play and choosing the location, your opponent remains the controller of the Thousand-Tailed Watcher as it enters the battlefield.
2. Referencing "Enemy": When a triggered ability uses the terms "enemy" or "friendly," these are checked from the perspective of the controller of that ability (your opponent).
3. Application: Because the Watcher is controlled by your opponent, its "enemy units" are your units. Therefore, when the ability resolves, your units receive the -3 Might penalty.

Sequence of Events:

1. You resolve Bone Skewer. You choose a battlefield and your opponent chooses Thousand-Tailed Watcher to be played there.
2. The Thousand-Tailed Watcher enters the battlefield under your opponent's control.
3. Its triggered ability ("When you play me, give enemy units -3 might...") is placed on the chain.
4. The chain resolves (LIFO). When this ability executes, it checks the battlefield for units controlled by your opponent's enemy (you) and applies the -3 Might reduction to them.