I recently returned from WDW and though my family did not win anything, we did see quite a few people walking around with prizes. We did not see anything actually "awarded" but we did come across several members of the Dream Squad in all of the parks.
Most of the awards seemed to be either the Mickey ears, the Park Fastpass or the pin lanyard. I spoke to one young man on BTRR who had recieved two awards. He told me that he got a pin lanyard after exiting a slide at Typhoon Lagoon and a Fastpass after exiting Star Tours. Another lady at EPCOT said that her family received Mickey ears after participating in the fire exhibit in Innoventions.
Almost all of the people I spoke with said that they received their prizes early in the day. We did, however, see members of the Dream Squad with the white vests on, walking though Innoventions in the early afternoon. They looked like they were just returning from an "award".
I overheard a women asking a supervisor in EPCOT (near MS and TT) about winning a prize. The supervisor explained to her that they do not know where they are going until 45 minutes before the "award".