If I gain control of my opponent's creature (a Ravenous Chupacabra, for instance), does that creature's "enter the battlefield" trigger happen again?
No. Taking control of your opponent's creature will not trigger an "enters the battlefield" ability.
The glossary section of the Comprehensive Rules defines "Enters the Battlefield" as follows (emphasis mine):
Enters the Battlefield
A nontoken permanent “enters the battlefield” when it’s moved onto the battlefield from another zone. A token “enters the battlefield” when it’s created. See rules 403.3, 603.6a, 603.6d, and 614.12.
In your example, the Ravenous Chupacabra was already on the battlefield. Changing its controller does not cause it to change zones (i.e. it never leaves the battlefield) so the creature does not "enter the battlefield."