Radagast and Gandalf came from Valinor, where they lived in their "free form" known under their old Quenya names of Aiwendil and Olórin. When they came to Middle Earth, they assumed a mortal form and took the names of Radagast and Gandalf. But surely, they would have recognized eachother when in Middle Earth. Why didn't they use their Quenya names? Also, I would assume the same for Saruman (originally Curumo), and even for Sauron, because Sauron would know all of the wizards from the time when they all lived in Valinor.
It's important to realise that Lord of the Rings is feigned to be a translation of Bilbo's and Frodo's memoirs (and Bilbo is an unreliable narrator to begin with). What you read in the book is therefore not an absolutely literal account of what was really said, and shouldn't be taken as such.
In the specific case of all dealings between Saruman, Gandalf and Radagast, with the exception of the Voice of Saruman scene, what you're getting is a fourth-hand account of proceedings. So you have an event that happens, then Gandalf relays it to the Hobbits, then the Hobbits record it in the Red Book, and then Tolkien translates from the Red Book and you read his translation.
In such a case, Gandalf is most definitely not going to use real names when relaying the events to the Hobbits; as he said himself in another context:
Olórin I was in the West that is forgotten, and only to those who are there shall I speak openly.
Note: Following Andres F. suggestion, here is a somewhat expanded repost of my comment as an answer, only focusing on Sauron case yet. If I find more elements than those already indicated here, I will add them later.
Sauron's name is based on the Quenya adjective saura which means "foul, putrid", while his original name was Mairon, "the admirable". In this case, it makes sense for Gandalf and Radagast, as fellow Maiar, to refers to him as Sauron rather than the original glorious one he lost any right to bear after turning to darkness.
In fact, the exact same thing happened to Melkor "He Who Arises In Might" who became known to everyone as Morgoth "Black Foe of the World" after his unspeakable acts agains the rest of Arda and its inhabitants.
Just for the sake of completness, even Olórin, Melkor, Mairon or Curumo are a rendering in Quenya of original Valarin names, which are in most case unknown since the language of the Ainur is very rarely used orally or in front of Elven listeners: Manwë "true" name is in fact something like Mânawenûz.
It's possible that they did, and we just don't know about it. As far as I remember there are no scenes between Gandalf and Radagast in The Hobbit book; in fact I don't think Radagast is mentioned at all. Any scenes between them in the movies (of which I've only seen the first so far) are inventions purely for the movie adaptation; given that a lot of the viewers won't be familiar with the entire lore surrounding the books it serves no purpose to confuse them by throwing in new names.
Even in The Lord of the Rings, I believe Radagast is only spoken about by Gandalf and Saruman, he doesn't actually appear as a character - Gandalf mentions that he'd spoken to Radagast before going to Isengard at the Council in Rivendell, for example, and I believe he's mentioned again when Gandalf comes to Isengard after the Battle at Helm's Deep - and given their mixed company it makes sense for them to use the name(s) that others would know.