Some horses need to go to a professional trainer (not like an instructor trainer, but like a serious, only works with horses, kind of thing. Like an obedience trainer for dogs.) This can happen when:
a) Young horse, needs a good start and the owner can work on polishing things up later, instead of risking the foundation work.
b) Problem horse that really needs to be whipped into shape, and the owner either doesn't have time or the knowledge on how to do it properly.
c) No time, owner fed up, miscellaneous reasons.
In my area, not too many horses go out for training. Usually owners will work with an "instructor trainer" to fix problems, and maybe once a week the "instructor trainer" will hop on to get the horse focused and minding its manners. Going out for training is a bit more serious business, and usually deals with severe problems like rearing, bucking, and just being plain hard to handle. This also happens sometimes when owners buy green or young horses (and thankfully) don't want to risk messing up on teaching the horse foundation work, and will send it out to a professional to get the basics down pat.
If I am wrong, someone please correct me, but that is why I have seen horses go out for professional training.
Oh and to answer your other questions - no, the owner typically does not go with the horse if it goes out for professional training, and there isn't a course at the end (as far as I know.) If you're working with an "instructor trainer" than you work right along with your horse, and the trainer teaches more to the whole partnership rather than just the horse.
Hope this helps!