Jisho

This is called "jukujikun". It's when a Japanese word that already existed gets assigned to a pair of kanji together. You probably already know a bunch of common ones, like 今日 (きょう), 大人 (おとな), or 田舎 (いなか).

Those all fit the same method of word-construction as 蝸牛, but perhaps 蝸牛 just calls more attention to itself.

And yes, the kanji 蝸 itself is listed with that kunyomi reading of the full word, but I think it's mostly just a technicality. As far as I can tell, it basically isn't used alone.

Leebo already told you the general phenomenon that this is based on, but to answer your question directly: "Which kanji makes which sound?" When there is the word 蝸牛 written anywhere, then neither make any sound on their own. There is only the compound that cannot be further divided.