Yes. It can be done...but there is some validity to the concerns. Having once been a network admin, it is remarkable what people are willing to share - essentially giving up their own privacy. It can be done. However, we are talking about two separate issues here. Resumes and on-line privacy.
1) On-line privacy can be managed by controlling what you post within social networking group bios/profiles, messageboard bios, instant chatting programs, meet-up/yahoo groups, etc. First, you don't have to join each and every service out there. Monster.com...LinkedIn.com....are o.k. Your local community might have something too. Consider using your first initial and last name. Think twice before giving any info along with that to tie you where and who you are. Addresses, phone numbers, other e-mail addresses, etc....and certainly any interests or hobbies...especially if you have strong opinions, controversial interests, or anything else that you would not share with family members, friends or coworkers. There are countless places where data collects on-line, whether you contribute to it or not, as various databases are going on-line...and information is being shared among them. You can try Googling your name to see what is out there. Then, you can take steps to clear that info. You can either e-mail requests to have your info remioved (follow the specific sites details) or log-in and pare down what you've originally shared. Keep the info as vague as possible or delete it. I'd also add...don't respond to spam...be careful clicking ads on websites...consider getting a software firewall...use disposable e-mails (yahoo, hotmail, etc.) and don't put details in the username that identify you.
2) Resumes, on the other hand, are there to provide information. Consider discussing what should be shared and what probably shouldn't be shared with someone who knows what to look for. Local communities may offer courses through community college, or the chamber of commerce. You might try calling or speaking with people in Human Resource Dept...to see what they have to say about what they look for. Libraries are good sources for books on the topic, but make sure they are current publishing dates. Libraries may offer workshops..or other local places. Ultimately, you can find out the best format to use to showcase your abilities and work history. You may or may not include dates (to avoid ageism - to some extent) and you don't necessarily offer information about marital status, family, religious or personal participation, etc.
Now, to merge these ideas....the balance is up to you. However, I am old school. The internet is very cool, but it's not a magical device. Blasting the internet with your data....having it bounce around to all sorts of places, including unintended one, isn't necessarily going to do a lot of good. The data, once it is out there, can work for you or against you. It is possible to use it selectively... You can use e-mail to target specific people at specific companies. Network with people and find out who you can ask questions of...or find out about positions in a company or sector. Word of mouth and in-person contacts are the way to stand out among the masses.
-just my 2 cents