I think 'breathe' is the wrong term. I aerate red wine simply by pouring it into a decancter (jug) a little while before serving. It does seem to make a difference.
It does make a difference if you let it breathe, i normally pour mine into a jug with lots of splashing (not carefully letting it run down the side of the jug) then pour it back into the bottle, have a mouthful before you do this and try it after, you will notice a difference.
Malcolm Gluck, wine critic for The Guardian, was all in favour of letting wines breathe, as in the link below. Despite the content of that article, I remember him being challenged on TV, when, after several tastings, he had to agreed that certain reds tasted better straight from the bottle. It's all up to the individual wine, I guess.
I don't know about current predominant belief, but I do know that some wines seem to benefit from doing so. In fact some wines I've had tasted better the half bottle left to the following day than the first opened glasses. I suspect it depends on the wine, but worth doing anyway.