I soak my beet pulp but not for long. I measure it out first, toss it into a bucket and add some water and let it sit while I am measuring out the feed for everyone, so it sits about 5-10 min. Just long enough to suck up some moisture, that's all it really needs.
I would think it would be kind of like that stupid "cinnamon challenge" that was going around on youtube a while back. People would swallow a spoonful of cinnamon and it would dry out their mouth, stick, and then when they went to cough, they would choke on it. So I was thinking, as quickly as beet pulp absorbs water, it is very possible that it could dry the mouth and throat out as soon as it's swallowed, causing coughing, maybe inhalation, and possible choke. Wouldn't happen to every horse, but could easily happen to some, so adding water is a good idea.
"Keep a leg on each side and your mind in the middle"