Does a creature entering the battlefield under your control trigger 'enters the battlefield' abilities if it was put there by an opponent's spell?

Yes, absolutely. 'Enters the battlefield' (ETB) abilities are triggered abilities that watch for a specific event: a permanent entering the battlefield under a particular player's control. They do not care about who caused that event to happen. This is established by CR 603.2, which states that triggered abilities trigger when the event described occurs, regardless of the source.

So if an opponent casts a spell like Bribery (which puts a creature from your library onto the battlefield under their control) or a spell that puts a creature onto the battlefield under your control, any ETB ability on that creature still triggers based on where the creature ends up — not who caused it to be there.

The key check is: whose control does the creature enter under, and does the ETB ability match that? Per CR 603.6, the game checks the state at the moment the trigger event occurs to see if the trigger condition is met. If the creature enters under your control, your ETB effects trigger; if it enters under an opponent's control, theirs do.

Concrete example: Your opponent casts Bribery and puts your Mulldrifter onto the battlefield under their control. Mulldrifter's ETB ability triggers and your opponent draws two cards — because Mulldrifter entered under their control, they control the trigger.

Bottom line: the origin of the spell or effect is irrelevant. ETB abilities trigger based on the entering event itself (CR 603.2), so any creature entering the battlefield under your control will trigger your ETB effects, even if an opponent's spell caused it.