Yes, they do. They need to show proof of having advertised the job somewhere. At least, I've been through this process three times, with three different lawyers, and we've done it every time.
Exhaustive discussion here: https://news.ycombinator.com/item?id=10969721
But note in particular that firms specialize in coaching companies on how to fake compliance with such laws:
story: http://www.post-gazette.com/business/businessnews/2007/06/21...
video proof: https://www.youtube.com/watch?v=TCbFEgFajGU
Neither of those jobs actually meet immigration categories. If you want to hire someone through H1B you need to provide a job spec that is needlessly specific with exact skills checked off. The kind of job that, if you saw it, you'd think "I never want to work there, they list a job as if it's a checklist of acronyms!"