I got exactely the same problem over here. Field types “Website” & “Birthday” don’t show up on the “edit profile” screen. They don’t display on the “view profile” screen even so. All I see is the message from the privacy plugin, telling the visibility for the specific field.
Other field types like “Textbox” or “Checkboxes” work well.
I am using WordPress 3.5.2, BuddyPress 1.7.3 and BuddyPress Xprofile Custom Fields Type 184.108.40.206 – my theme is Custom Community.