You definitively can! I just “met” two other Spits very spontaneously over the channel, near the Cliffs of Dover (seems to be a good “meeting place” for Spits ;)).
And I could even see their distinct liveries (they had a different livery selected than I - of course I am talking about the default Spitfire liveries here, but it should also work for “custom liveries”, for as long as everyone has them locally installed, too).
Watch out for “SPIT” in the player name plate: if it says “SPIT” then it is a Spit 
“Proof”: Steeler's Photo & Video Diary - #19 by Steeler2340
P.S. If you don’t see any aircraft short name at all (like A320 or SPIT etc.) then I guess the other person is using an aircraft which you do not have installed
UPDATE: Oh, and make sure that you have “generic multiplayer airplanes” (?) disabled! There is a setting (in the Graphics section? Or Data?) where you can choose - for multiplayer and AI aircraft - whether simplified “generic” models should be used, or the “real” aircraft (geometry model).