No, an employer cannot get access to social security records or W-2 forms, and no, certain employers do not have more rights than others. However, it is by no means impossible for an employer to find out about jobs you have not listed if they really want to, particularly if the industry/position is a relatively closed one. It is also entirely possible (I've seen it happen many times) for an employer to receive information about an employee without even looking for it. Examples (keep in mind that each and every one of these examples really happened to me):
1.) Someone I worked with at ABC company got a job with XYZ company, which is where I was working at the time
2.) Someone I worked with at DEF company became a recruiter, and was called on by one of my hiring managers when I was working at TUV company (I walked into the kitchen and he was having a cup of coffee while he waited for my manager to get in)
3.) A co-worker of mine quit her job at ABC company and got one with the insurance broker DEF company was using when I started there (which is how I found out not only that she'd been fired, but why)
4.) My boss at XYZ company quit to go work for a competitor. Later XYZ merged with GHI company to form JKL company. I stayed with XYZ through the merger. My former boss quit the competitor to take a job with JKL company, and thus became my boss again.
5.) My boss in 4.) above, recruited several of his team to leave XYZ and come work for him at the competitor. Several of them also quit the competitor to work for him at JKL, thus becoming my co-worker again.
6.) When DEF closed and I went to work for MNO briefly, MNO was using the services of JKL, and my contact person there was someone I had trained while I worked there.
In each of these situations, either someone could easily have told my employer about any jobs I left off my application, or someone could have told me, as the HR manager, about the job history of someone else. BTW, while I'm sure following all these initials is confusing, all of them except JKL and MKO were the same industry, and it's a HUGE industry. MNO is also a very large industry, and JKL is one of the fastest growing industries in the US. All three industries employ literally millions of people; hundreds of thousands just in my fairly small state. So even though it's quite coincidental that even one of these examples happened, all six of them really did.
The bottom line is that even if an employer doesn't go looking for additional information on you, you can't count on him not receiving it.
