Ewww, you do not want to eat the skin, at least on a full-grown beet! If you get little baby ones - like golf ball size - you could, but I wouldn't.
We grow beets, and I love them in all stages, but I've never heard of that salad. I'm going to try it. I love them best when they're little. I boil them, peel them (as described above) and eat hot with just a litle real butter and salt. Have almost a corn-like flavor. I can my own pickled beets, and we eat them all winter, either as a side or an addition to a salad. Roasting is good for the bigger beets. Like all roasted veggies, they get sweeter.