Wet diapers feel warm and wet (unsurprisingly). They also feel soft and squishy compared to dry diapers - dry diapers are quite hard and you can't really compress the padding with your fingers, whereas you can sort of squidge wet padding. They're also heavier - the padding gets heavier as it gets wetter, so it pulls away from your body and hangs between your legs.
Messy diapers I know less about. I know that messing a diaper willingly is really difficult because I'm toilet-trained and my toilet training says it's absolutely not OK to do that kind of thing - I don't feel any emotional problems doing it, but making the muscles do the right motion without being on a toilet is really hard. It's easier with Dulcolax, which makes your butt burn, and doesn't so much make you lose control as makes you have to concentrate very hard on retaining it. I haven't tried castor oil yet, but I've heard that gives you one big, painful accident and then painless but uncontrollable voiding for anywhere from 9 to 48 hours.
The actual sensation of messing is interesting. Your diaper does get a lot heavier. The sensation of messing while standing up is quite pleasant, although how you move will affect how much self-cleaning you have to do afterward - if you sit down in your messy diaper, you'll have mess spread over your entire diapered butt, as opposed to it mostly being a lump in one place in the seat. You'll feel more safe messing if you have a diaper which naturally leaves space under the butt when fastened correctly, like a Bambino Bellissimo. Honestly, being in a messy diaper feels a lot better than you'd expect it to.
I don't know anything about masturbating in messy, or wet and messy, diapers, but I know masturbating in a wet diaper is a lot more fun than doing it almost any other way - the warm wet feeling is divine, and you don't have to clean up afterwards. I feel the need to get out of it afterwards, but that's only because I become more aware of how noisy the diaper is and how I might be walked in on; I don't actually feel any revulsion toward the diaper itself.