Feature request: File CV anonymously, or with some personal data excluded. Then I can only reveal myself to potential employers after getting relevant inquiries.
That's an interesting idea, but if I were an employer I would have a hard time justifying paying for a service where I can only see a small bit of anonymous information. You'd have to unlink the stackoverflow account to really be anonymous, which would eliminate a huge feature of the SO CV search service. Even if you showed a rounded off reputation score, employers still wouldn't be able to review your questions and answers.
In another sense, it's like looking at advertisements which give only a little information, but force you to contact them for details. "If you have to ask, you can't afford it" is one perception, the other is that when compared to a search that's full of other candidates who have their information open and available your 'closed' listing looks less attractive.
Hiding form your current employer, however, seems the easiest to resolve, though. If this CV is 'discovered' then the response is simply, "I keep my resume updated and online regardless of my long term plans. It's not an intention to leave my current job, it's a pro-active realization that even though I plan on continuing with the company, and even though the company plans on employing me long term, sudden economic shifts may affect both of us. If anything, it's simply being as prepared as one can be for the unpredictable."
Different answer because it's a fundamentally different approach to the question.
How is the current system not anonymous?
Check it out:
You are literally not required to enter ANY information in any field EXCEPT an email address when filing your CV (which you would need for careers to contact you on a hit anyway). Your CV is separate from the billing system, so you can enter (or not enter) anything in any field and it won't be associated with whatever you put in the billing information, which won't be shown to employers.
If you require them to contact you prior to gaining access, then why put your sensitive information in the CV at all? It won't ever be visible to anyone.
Even if you wanted to show up in location specific, or skill-set searches, but make your location/skills unavailable unless contacted, they can run a series of searches to find out where you live or what your skill set is, so there's no point in anonymizing that but also making it available for searches.
As far as I can tell, you can be as anonymous or open as you desire, and require them to contact you for more information.
Please try out the system more fully and describe exactly what you want to make anonymous, but can't currently, or how the anonymous experience could be improved. Right now you are making a very general request, but as far as I can tell one can immediately fulfill your request right now in the system as it is currently implemented. You need to be more specific. Examples would be very helpful.