They just separate the sounds to spell things out. I remember a scene from an anime that worked like this. I realize a lot of people are down on anime, but I can't see how that would affect the validity:
"My name is あんず"
"I'm sorry, I didn't catch that."
"あんず! Don't you remember me?"
"I still didn't catch that."
"あ! ん! ず!"
Like she were saying separate words.
Japan *does* have kanji contests, but they don't work like spelling bees, since they are necessarily written. There are even kanji exams that you can take for bragging rights. One of the recent prime ministers made a lot of Bush-like gaffes with reading kanji wrong while giving prompted speeches. The number of citizens taking the kanji exams skyrocketed afterwards.
Kanji kentei - Wikipedia, the free encyclopedia
If you're (for example) talking over the phone, and need to explain the kanji of a word to someone else on the other line (such as your name), since almost every kanji has a kunyomi as well as an onyomi, you can usually give the other pronunciation to help out. "That's もと as in ほん" for 本. Otherwise you can give a word that begins with the right kanji, just like we say "R like Roger." "意 as in 意味." Finally you can describe the parts of the character itself. "弧 with 弓 on the left, not 孤 with 子." (Although that last one would probably require an additional 子 like 子供)
In person, you can write the kanji in the air or on your palm with your finger, which is just faster sometimes. Be sure to use the right stroke order.