The Bucks sent the Nets two second-round picks (one in 2015 and another in '19) for Kidd, who went 44-38 in his only season as Brooklyn coach.
"When you list the characteristics that make a successful head coach, you would include leadership, communication and a competitive drive," general manager John Hammond said in a statement. "Jason used all of those traits to become a 10-time All-Star player in the NBA, and has now translated his on-court success to the bench. We welcome him to the Bucks organization and look forward to building a Championship-caliber team with him as our head coach."
Milwaukee fired Larry Drew on Monday after reaching the deal for Kidd. Drew went 15-67, the worst record in the league, in his only season guiding the Bucks.
"Despite the challenging season, Larry always handled himself and represented the Bucks in a first-class manner,'' Hammond said in a statement on Drew's firing Monday. "Larry did the best he could in a difficult situation, especially given all of our injuries. I want to thank Larry for all of his efforts, and we wish him the best of luck in his future endeavors.''
Bucks co-owners Marc Lasry and Wes Edens said in a statement that Kidd "is a determined leader, a tough-minded competitor and a great teammate."
Kidd sought control of the Nets' basketball operations department after his first season as coach and was denied. The Nets then gave him permission to talk to other teams about a job.
It was a stunningly quick ending to Kidd's reunion with the franchise he twice led to the NBA Finals as a player. The Nets hired him last June as coach just weeks after he retired as a player and retired his No. 5 before a preseason game in October. Also, he bought a small portion of the team.
A league source told ESPNNewYork.com that Lionel Hollins has already emerged as "a very serious candidate" to become the Nets' head coach.
Nets general manager Billy King said Tuesday that the team's brass had dinner with Hollins on Monday. The team met with the former Grizzlies coach again Tuesday.
Asked during an interview with WFAN later Tuesday, King said his meetings with Hollins "went well."
"We met again today, and the meetings went well," King said. "So now, after I finish [this interview], I'll meet with the staff, talk to ownership and then look at our list and see if we have to go further because we've got a list that we'd like to go through if we need to."
The Nets would like to have a coach in place sooner than later, however, because the free agency period has already begun.
Hollins led the Grizzlies to three consecutive playoff appearances in his last three seasons in Memphis. In 2012-13, his last year at the helm, the Grizzlies won 56 games and advanced all the way to the Western Conference finals. The two sides ultimately decided to part ways and Hollins' contract was not renewed, apparently because there were "philosophical differences" between Hollins and ownership.
"If you look at track record and what he did in Memphis, they got consistently got better every year," King said.
Ettore Messina, former coach of CSKA Moscow, also is a leading contender for the Nets' opening, a source told ESPN.com.
ESPNNewYork.com's Mike Mazzeo, ESPN The Magazine's Chris Broussard and The Associated Press contributed to this report.