Your HR sees your resume
HR here. It depends. I usually throw them away. However,at one company that I was at, they (not me) use to find the originals in the copy machine. Those employees were given the opportunity to use their resumes sooner than they expected.
As a rule, if you are job hunting keep your resumes at home and keep your mouth shut at work. Just a little insider tip.
I have read articles that only a small percent of employers take the drastic measure of termination if they find their employee's resume online. Most understand that it is to be expected or evaluate the employee's work performance and work environment.
Most boards give the option of posting without your name and contact info. Do that if your employer wouldn't approve.
If the policy of a company is to fire employees they stumble on to looking elsewhere, that employee is probably better off not working there and the employer shouldn't be surprised people are looking to leave.
Boards do offer the option of making your contact info private but there are plenty of other information that can give away your identity if the HR remembers you. It depends on the situation of course, most HR run through hundreds of resumes and is not likely to see something in yours that makes them compare the one they have on record to the one they found online. It is still a possibility I like to avoid if possible.
If your resume is posted on a website, job board, etc and you insist on supplying your name and contact information, then you run the risk of exposing your job search to a very large number of people who probably don't have your best interests at heart. Use a lot of care and caution coupled with an extra dose of common sense.
Most employers are well aware that today's job markets are much more fluid that in years gone by. Employees and employers are both expected to keep options open for better jobs/careers and better employees. Reprisal is unlikely but be ready to talk if HR or your supervisor queries you about it. You never know, it could be an opportunity to pursue something better at your current work.
I think it definitely depends on your company. In any case, it would be best to keep your contact information hidden if possible.
A little over a year ago, my boss told me during a one-on-one that our HR found a posted resume on monster.com of someone I supervised. I wasn't sure what I was supposed to do about it, so I asked. He said that there wasn't really much that either of us could do. This particular employee had been waiting months for a promotion, so we knew they were getting impatient. Their promotion was up to upper management above both me and my boss, so I wasn't really even sure why he gave me the warning. There was nothing I could do, and my boss and I knew we couldn't exactly approach the employee and say something about it - that would just be wrong unless we could offer them something.
I would hope that if any company's HR spotted the resume of their own employee, they would at the worst, do nothing. Hopefully, they'd try to get the person to stay.
I only hope that most remain professional about it though.