who is this third party that does this anonymous transactions?
Someone who set's up a mixing node, which consist of 2 different servers that talk to each other (preferably in separate countries). There is a transaction fee for using anonymous transactions, the node gets these transaction fees. People say its centralized, however once you get numerous people using the anonymous transaction feature you are going to have a lot of people that setup nodes to collect fees as an alternative to mining.
Same as TOR with entrance and exit nodes, only that unfortunately in this case entrance and exit nodes are the same?
Couldnt they just mimic TOR's system where one mixing node sends the coins to the next and so forth in order to get a couple of them between you and your destination?