I tagged all my contacts on my Picasa Web Albums, but I was wondering if this data is stored in the photo itself or would I have to retag every picture when I import them to my desktop or share them with someone else?
Coming from the reverse, in the Picasa Desktop app tagging the photos does not change the photos at all (I know this as mine are never changed in my incremental backups after tagging).
The face information is stored in the picasa.ini file linked to a database file in the user profile.
When I have uploaded the pictures to Picasa Web the face tags are replicated there. (once I switched on the settings)
I just tried using Picasa Web's Download to Picasa function on a second PC and all of my face tags were lost.
So I think the answer is, unfortunately, you will have to re-tag.
Suggestion: Use Picasa Desktop and it's Sync To Web function to keep your Face Tags in both locations.