someone who has passed away.  I have heard 3 months, right away, one year.  Does it depend on the Medium?

Generally speaking - 3 months to settle has been the norm.  Of course, some of us have had communication much sooner and it was quite valid.  

I was working in a church one time and a young man, full of lively smiles, came to a lady.  He had passed in a motorcycle accident just 2 weeks prior.  It definitely was a strong connection.

Also, most spirits wait around the earth, before going into the light, until at least after their funeral.  They tend to want to see the gathering.  I had an aunt that appeared to me and said - I knew I would draw a crowd.  She was (is) such a fun person.