Hooves are made of the same stuff that finger nails are, and have no nerve endings. The most they can feel during the nailing process is a little bit of pressure.
In regards to sneaking, any hooved creature will have a penalty to Move Silently on hard surfaces, with or without being shod.
Up until the point that someone screws up a shoe... Oh and then all hell breaks loose.
I am not an equestrian. I worked on a horse farm for years, shoveling manure, tending horses, currying, walking, lunging, rasping the occasional hoof while also cleaning the hoof, giving massages (yeah, apparently people assume I know how to give massage to any species because I can massage people?

), racking hay, pouring grain. I've seen horses get hot-hoof/laminitis leading to founder. I've seen horses that operate quite well with riders and NO shoes. Most 'skilled' farriers (the guys who shoe horses) are starting to move away from the horseshoes of old and into more temporary/no-nail secured shoes. From horse boots (once used for orthopedic issues in lieu of corrective horseshoes) to the strict 'no shoe' groups, most horses don't need horseshoes.
Maybe you remember going on summer vacation or just spending time at home during the summer if you were lucky like me and occasionally lived in a warm clime

. You walk around, your feet will develop calluses, and you develop strength in the foot.
A horse's hoof is just a complex version of your own nail attached to the phalanges. They bear their weight on the legs and, if left to the wild or nice variant but non-soft surfaces will allow for a shoeless horse. Now horses should have protection over certain surfaces (sharp pebbles for example) to protect the frog and other pieces of the foot. Some of the illnesses of the hoof (abscesses, laminitis, thrush, white line) are caused by poor conditions in the stall or generally poor conditions and lack of movement in cold and/or wet bedding/ground that needs to be cleaned.
Now let us think of a sentient creature who feels that pain and can state it (or even take care of it themselves). For two-legged hoofers? Hell, that is an easy one, cleaning themselves just like I would with a horse. A scraper, clean it out, apply appropriate substances, relax.
For note yes, use horseshoes for the bellicose, or those who live in areas or cannot easily take care of their hoof. If they can? They're probably using something along the lines of this:
There are versions of this allegedly predating the shoe. You can get better tractions through hobnailing the boots, and allows for the floors not to get scuffed up by giant pieces of metal dragging across them.
Of course, my biggest question about a Centaur is how do they wipe...
Slainte,
-Loonook.