Which side decides which variation of the Sicilian the game goes into?

I see WAY TOO MANY people make this mistake of deciding on openings based on who they perceive to be the one that chooses the opening.  EVERY GAME is decided by BOTH players.  Just because one side's move determined the name doesn't mean jack.

Example:  White play 1.b4.  Yes, this is the Sokolsky.  Just because White's first move caused the naming of it means nothing.  1.b4 e5 is vastly different than 1.b4 d5 which is vastly different than 1.b4 c6 which is vastly different than 1.b4 Nf6.

In the French Defense, 1.e4 e6 2.d4 d5 3.Nd2, White's third move is what causes it to be called the "Tarrasch" Variation, but you have the Open Tarrasch (3...c5), Closed Tarrasch (3...Nf6), Guimard (3...Nc6) and also 3...dxe4, which transposes to the Rubinstein Variation.

I always tell people to forget about the names of openings.  All they are good for is identity when titling books, like whether a book covers the French, Russian, Pirc, or Sicilian Defense.  Otherwise, names are useless.  Understanding the position is the criticality.

I respond to 1.e4 with 1...e6.  I could care less if this was called the French Defense, like it is, or if it was the Persian Defense, Australian Defense, Iranian Defense, Jupiter Defense, Neptune Defense, or Diarreah Defense, it wouldn't matter!  What matters is that I understand the purpose of 1...e6 (to prepare 2...d5 without having to recapture with the Queen if White were to take), the benefits (being able to play ...c7-c5 in one move to attack White's center if he advances e5 at any point, unlike the Caro-Kann, which requires 2 moves of the c-pawn to break the White center, and lastly the consequences, namely in this case the bad bishop on c8.  You need to know the ideas behind the opening, like the typical Queenside attack in the Winawer, down the f-file in the closed Tarrasch, Queenside in the Advance, and then understanding the symmetrical positions from the Exchange.  Simply knowing that it's called the French Defense, or which player is the root cause of it being a French, are both totally useless and not worth your time trying to figure out!

You're right in the sense that the "name" of the opening changes depending on the move. Let's say we want to play the Sveshnikov:

Each of the "branches" has a different name, and there are more variation/sidelines I haven't mentioned. You can relate this to a causal political discussion: Two player take turns to define the opening they're both playing. White plays 1.e4 and decides it's a King's Pawn opening. Black playes 1...c5 and it's a Sicilian.  White plays 2.Nf3 and 3.d4, making it an Open Sicilian. Black plays 3...cxd4, 4...Nf6 and 5...e5, entering the Sveshnikov Variation.

In any opening, be it the Sicilian, the Pirc or even the Grob, both sides have a few choices to make. To study an opening means to get prepared for all the choices one might encounter in an actual game.

The answer is BOTH SIDES.

It hinges on both sides' decisions.

Using your Pirc example:

1.e4 d6 2.d4 Nf6 3.Nc3

-- White can also play 3.f3, avoiding the Pirc - I do this - and Black can head into a Saemisch Benoni (3...c5 4.d5), Passive Old Indian setup (3...e5 4.d5 with c4 coming), or a Saemisch Kings Indian (3...g6 4.c4 Bg7 5.Nc3)

3...g6

-- Black can also play the Pribyl, which is 3...c6 and if 4.f4, then 4...Qa5 5.Bd3 e5

4.f4

-- White can also play 4.Nf3 or 4.Be3 - against the latter, Black has 4...c6 or 4...Bg7

4...Bg7 5.Nf3 and now it is Black that chooses between 5...c5 (against which there is 6.Bb5+ or 6.dxc5) and 5...O-O (against which there is 6.Be2, 6.Be3, and 6.Bd3).

So no, white does not call all the shots!