This is from The Writer's Complete Fantasy Reference compiled by Writer's Digest, p. 164.
Merfolk
"Water-breathing races have appeared in a variety of legends. The Greeks had their sea nymphs, the Nereids. Triton, the son of Poseidon and the Nereid Amphitrite, had the tail of a fish. In fact, his name became a generic term to describe men with the body of a man and the tail of a fish. More commonly, these legends developed into the tales of mermaids and mermen, whose chief characteristics as a race is their half-fish forms and their ability to live underwater but not on land. Mermaids sometimes had a reputation for mischief if not disaster. Though some mermaids gave up their tails to join men on land, there are also stories of mermaids dragging mortal men beneath the waves to join them. Other times, mermaids caused shipwrecks, especially if one had fallen in love with a sailor. Hans Christian Anderson's "Little Mermaid" is probably the most famous tale of merfolk. In his classic story, the little mermaid trades her voice for legs and the chance to win a mortal husband. Of course, the original tale does not have the happy ending the Disney version put on it."
So according to this they breathe water. And unless they have magic performed on them, they can not live on land. However, I am currently re-reading Piers Anthony's Xanth series and his Siren is a mermaid who has the ability to trade her fin for legs at her whim.
Many stories show mermaids partially in water and partially out which would make me think their gills are in their tails, but I have never really read it mentioned particularly. Then they are often seen lounging on reefs or large rocks in lakes. This to me would seem to mean that they can breathe both in air and water.
I am not sure you can "go against the grain" with merfolk. Heck, maybe I just confused the issue more lol.