No, not rear. In equine circles that's technically called "the rear".
At least in the UK, Ireland, Oz etc, the near side is the side you get on, nearest to the kerb -- in other words, the left. The right side is the off-side. Cars and other vehicles are often labelled like this too.
I haven't a clue whether off and near are reversed in countries where they drive on the wrong side of the road.
However, I think that in all countries horses are normally approached, tacked, led, mounted and handled from the left side, so perhaps it's the same everywhere. Any Americans out there...?
Cattle (or at any rate dairy cattle) are traditionally milked from the right side, and so the near side of a cow can sometimes be the right... We don't milk most of our own cows, but when we do it's the near-est side, whichever that happens to be.
When driving horses, there are special words to tell the horse which way to turn. These vary from place to place, but in UK the commonest are ree for right and haw for left (as well as gee for forward and whoa for stop). Does anyone know any others?
When being driven, my cob listens for the word "trot", and will do it instantly if I say it in a sentence by mistake. I have to spell out T - R - O - T, like a dog with W - A - L - K. I'm hoping she doesn't learn to spell...