How does the Mangekyo Sharingan blind its wielder, and why are some wielders not blinded?
An original Mangekyo Sharingan puts a strain on the eye. The ocular muscles are placed under a constant strain due to the unique chakra the Mangekyo consumes. That strain causes the user to gradually go blind.

Another person's eye doesn't (for some reason) produce the same strain. So the user's vision is intact.

Let's review the list:


Itachi - Awakened his Mangekyo on his own, going blind.
Sasuke - Awakened his Mangekyo on his own, gone almost completely blind, then took Itachi's eyes, and got his vision back.
Madara - Same as Sasuke.
Kakashi - Remember that the eye didn't originally belong to him. His Mangekyo is more similar to an Eternal than a normal one.
Obito - Presumably, it's because Obito has a high concentration of Hashirama Senju's cells. These constantly heal the eye, preventing it from going blind.
Danzo - Same as Kakashi, the eye didn't belong to him. Although using Sishui's eye did put a huge stress on his chakra and body.