After sending them a resume, LinkedIn contact information, Twitter, Github account etc. I was hoping for mutual respect from the AWeber HR person to verify who they were via a 3rd party (LinkedIn, Twitter, Google+, Github... Oauth).
Agreed. This idea of verification is probably new to employers. And the company has claimed that they are, who they say they are. Plus, they have provided the LinkedIn URL.
But, if I were a tech company, I'd realize the advantage of making 3rd party verification an integral part of how your company deals with the public -- when requested.