I played in both the US Open and British Open last year and received the following:
USGA - 1 year standard membership package, rule book, bag tag, membership card and blue hat from them.
British Open - hat from St Andrews.
I believe that you had to not only qualify but play in the tournament to receive the prizes. For the USGA I was contacted by them and had to give them my information so that they could register me as a USGA member and send the prize to my home, I believe the same was true for the British. It took several months to receive these, I think July / August for the USGA and a few weeks longer for the British.
I have not seen if the prize for this year is just a USGA one year membership and the hat or not. I also know that shipping to various countries is costly, I believe that last year the British batch shipped it and then it was distributed from one US location.