Legit job offer?

The company might be legit, but the ad or the person responding might not be: a company I used to work for was often used in this way, to the extent they put a note about potential job offers on its website - see here under "recruitment scams and hoax offers".

If the company exists, try calling them up and asking to speak to the person you've been dealing with - don't use the number they've asked you to call on, but go through the main switchboard.

I'd do this before sending anything over....