Possibly Sketchy Job Opportunity?

Long story short, I had an interview that went really well, and the interviewer's business partner shortly thereafter called me to ask for my D.O.B. and SIN to put me on tax forms.

This raised a red flag to me since it seemed like a pretty classic setup for a situation in which I am the idiot for just handing out sensitive information over the phone.

I told them I would happily give them the info if they were on officially signed forms. I have been out of work for 3 months and am pretty desperate for income at this point, and would be pretty sour if there isn't anything sketchy about that and I had cost myself an opportunity by being unnecessarily weary. Advice and insight would be appreciated since, out of dozens of jobs, I don't think I've had many, if any, encounters quite like this when signing on.