Before a crowd of about 100 people, Mitchell was presented with his medal by 1999 winner and former Hawthorn teammate Shane Crawford.
Richmond skipper Cotchin was crowned with the game's highest individual honour by triple Brownlow medallist Ian Stewart.
"It's obviously different, something no one else has really been through, so Trent and I will share something unique for the rest of our lives," Mitchell said.
"I'll enjoy the moment. I've tried to be really humble in the whole thing, and obviously the decisions have been out of our hands so we just went with the flow," Cotchin said.