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Ghia it is possible ;D
Take a look on profile 1.65 on myxoops.org site.
It's alpha, but like other XOOPS module are highly customizable.
As I remember you can use $isOwner==1 and set permission for webmasters or admininistrator group to that fields .... you get what you want!
And there you will find extra administration for visibility. There U can set default permission ... IT SHOULD WORK.
PS Alfred (greet you cordially) said beta or stable will be realised soon (posted in november) ;)