It's the most annoying thing in the world. I have an hyphenated surname and it's really an annoying thing. 
The way I manage this is to pretend I have only one surname unless both are really required. Thus, both surnames only appear in official documentation like government records and the like. 
I'd say the default stance of the people who are presented with both surnames will be to say both, rather than one of the two. However, since I pretty much only use one, they will adjust over time and use that one.