Naming/describing kanji

When the Japanese need to unambiguously identify a kanji in speech (for example, when telling someone how to write something over the telephone), how is this done? Do people just think of a common and unambiguous compound or expression and explain that they are referring to, for example, the first kanji in that compound? Or do they describe the actual shape of the character in terms of radicals and stuff?

Both, depending. For example, one of my teachers described her name as 節子、「季節」の「節」に「子供」の「子」と書きます。Or if it's a common word you can say something like 「見る」という（の）字. Then there's the 「たけのこ」はたけかんみりに「下旬」の旬 approach, where you name the 部首 and then name the part that's a full character by giving a word it's used in. たけのこ, by the way, is 筍 (⺮ is 竹, and it's called たけかんむり).

What always amazed me was that no one ever blinked when the kanji used as examples in how something was spelled didn't have the same readings as in the required word in question. They just took it on face value as "kanji have many whacky readings" and rolled with it. Of course, this came up most often in names, and names are notorious for having kanji that are read in ways that are non-standard.