\Yes they can eat sugar beet as pulp or fresh. In fact it is a constituent of some pig feeds. I've given shreds to mine with their feed and it doesn't need soaking.
But, shreds or pellets are not an efficient feed to give to fattening pigs as they don't have the capacity to break down the fibre in the way that ruminants can. It is more a filler. I gave it to mine as I didn't want them to get any fatter, and because of the fibre content it has a similar effect to giving All Bran, in that it bulks up the droppings.
Fresh sugar beet is like giving them sweets. It gives them something to chew on and occupies them for a while. But I'm not sure how large amounts would affect them as it's pretty concentrated so I wouldn't give more than one a day each. I found mine really enjoyed carrots or fodder beet. We used to feed them those ad lib, as we had them for the cattle anyway, and again it filled them without making them fatter. (At this stage my mini pigs were full grown, and I hadn't yet decided what their fate was to be.)