Wednesday announced last month that the pair, along with Massimo Luongo, had rejected offers to remain at the club.
The duo have now been snapped up by Derby, whose future was secured last week following a successful takeover by Clowes Developments.
The club have got to work on their recruitment with the ex-Owls pair opting to join the Pride Park club, who will face Wednesday twice in league competition next term - including a potentially mouth-watering clash at Hillsborough on the final day.
Liam Rosenior has been given the reins on a temporary basis at the Rams after Wayne Rooney left his role as manager.
Mendez-Laing, who joined the Owls in November, made 19 appearances during his short spell with the club.
"I think it’s been in the shadows for a while now and I’m glad the club has got its situation sorted so I could finally get it done," he said following his arrival at Pride Park.
"I’ve played against Derby many times in my career. I’ve had a chat with Liam Rosenior and the way he wants to do things. It’s a massive club and it’s a project that I want to be a part of."
Wildsmith signed his first professional contract with Wednesday in 2013 and he went on to make 88 appearances in all competitions for club.
"It’s the first time I’ve been in this situation where I’ve not been at a club and out of contract," he said.
"So to get something done so soon after the last contract finished, and being at a club like Derby, I feel like it’s a great opportunity to come in here, enjoy my football and help this team get back to where it needs to be.
"I’m 26 and had been at Sheffield Wednesday since I was 12-years-old and you are going into a place where you’ve gone into work that long with the same four walls.
"I just felt like I needed a different challenge and I’ve got that in signing for Derby."