If you want the other person to see your custom form, they must have access to the form definition. In your scenario that means you need to embed it in each item. I had expected that to happen automatically, based on your description, but lets add one more setting -- on the Properties page of the form in design mode, check the box for "Send form definition with item." Then go through the same process of sending a rich-text message to a rich-text recipient with the attached item.
What version and build number of Outlook is the recipient using? If it's Outlook 2003 SP2, you may need to save the contact as an .oft file, then send that .oft with instructions for the user to save it as a file, then open it with Tools | Forms | Choose Form.
As you might have guessed, custom forms were never intended to be used in the way you're trying to use yours.
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Sue Mosher, Outlook MVP
Author of Configuring Microsoft Outlook 2003
and Microsoft Outlook Programming - Jumpstart for
Administrators, Power Users, and Developers