As far as I'm concerned, the tonality/atonality split is a red herring.
First, consider the differences, if any, perceived by a listener between a
highly chromatic score - say, by Ives, Bartok, Toch, or the Copland of the
Piano Variations - and a dodecaphonic work by Schoenberg or Berg.  If I
didn't know who wrote the pieces, I probably couldn't tell without a score,
and I doubt most professionals could, either.  Then there are those works
that are indeed dodecaphonic, but don't particularly sound so - parts of
Eisler's Deutsche Sinfonie, Thomson's Solemn Music, Ginastera's Piano
Concerto No.  1, Copland's Piano Quartet, Dallapiccola's Songs of
Captivity.  Is late Carter dodecaphonic?

Do you have to know something about the techniques of dodecaphonic
music in order to appreciate it? Knowing such techniques can enhance your
appreciation.  In my own case, I learned the techniques after I liked the
music.  I listened "naively" and enjoyed myself.  I learned something about
the construction and enjoyed myself even more.  How does this differ from
most classical music lovers' experience with tonal works? I don't see that
there is a difference.

As to whether you're dealing with real music if you can only appreciate it
through score analysis, it depends on your definition of real music and on
a lot of other circumstances, such as the quality of performance.  I doubt
Beethoven's Ninth was "real music" in this contrary sense for at least a
decade after it premiered, since that was when, by contemporary consensus,
the first decent performance (by the Paris Conservatoire, where the
performers had rehearsed it for more than a year) was given.

Steve Schwartz