wiki says it’s located in the “Palazzo di Spagna, seat of the Spanish embassy among the Holy See” but there’s pretty much no information other than that. I’m afraid it’s behind some closed door sort of thing and that sucks because it’s a great Bernini piece and I love his work a lot. Anyone have any clue?
The Spanish Embassy to the Holy See is actually in Rome proper, not in the Vatican. It’s in the Spanish Plaza at the bottom of the Spanish Steps - a famous location in Rome named for the fact that the embassy is there.
According to this website - http://www.embassypages.com/missions/embassy13782/ - it has office hours. Of course, I have no idea if the busts (apparently the Blessed Soul is there, too - Blessed Soul (Bernini) - Wikipedia) are viewable, but if any fellow dopers who speak Spanish want to check out the mission’s website, they might be able to tell. Or, if we have any dopers in Rome, they may be able to just walk over and check 
I’m also a huge Bernini fan, so next time I’m in Rome, I’ll definitely go and see.