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You didn't do anything wrong … unfortunately, that's how it works with all email programs I've seen.
Picasa stores the caption with the picture's data, but it's up to the viewing program whether or not you see it. If your recipient saves the email attached picture to their computer and then views it with Picasa – they will see the caption. But their email program does not display it. It's a real drag.
There is a workaround. You can 'stamp' the caption right onto the picture itself with Picasa's text tool. Click the text tool, then click the button that reads, 'Copy Caption'. The caption will now be text right on the picture. Of course, it may be hard to read based on the colors – but it's there.
Hope that helps.