Mouth radical on the left — the indexing radical. Marks 喊 as a vocal-action verb, joining 叫 (call), 喝 (yell), 嚷 (shout), 呼 (call out) in the family of vocalisations. The mouth is doing the work, with no other body part involved.
Right component 咸 supplies the sound (xián → hǎn, with palatal-to-velar drift seen in 减 jiǎn and 喊 hǎn). Purely phonetic — the 'all/everyone' or 'salty' meanings of 咸 don't carry into 喊. The phonetic series 咸→减/喊 is regular once you know to expect the j-/h- alternation.
