There is already a system in place for employers, by using their federal employer number and a password, to access Social Security information over the Internet in order to verify Social Security data. And no, a thief can't simply call the Social Security Administration to get personal data on others. The SSA doesn't give out personal data to anybody without first verifying that the person making the request has the need and the right to acquire that data.
If you have to question each document then there is no point in issuing those doucments.
It is an employer's responsibility to question each document in order to verify that the job applicant is legally allowed to work in the U.S. Unfortunately, many employers don't carry out their responsibility in regard to verifying ID and documents from job applicants.