Shumway met with head coach Wayne Gretzky in Los Angeles on Monday, saying the two talked about where the NHL team is headed.
"For instance, players will say they want to be in Phoenix," Shumway told the Arizona Republic. "Sometimes, I think that means the weather is good, the golf is good, and there are nice clubs in Scottsdale.
"What they mean is they want to retire in Phoenix. What we need are guys who want to play hockey in Phoenix. I don't think the nature of the city affects just us. It affects other sports teams in town, too. You get guys here that realize it's a great place to live and a great place to collect a paycheque. We need to get rid of that attitude."
The Coyotes finished last in the Western Conference and 29th in the 30-team league with 67 points, missing the playoffs for the fourth year in a row. The previous season, Gretzky's first as head coach, they had 81 points.
Shumway didn't point a finger directly at Gretzky for the team's plight. Gretzky owns a share of the team and is its managing partner.
"I can tell you that, over the course of the past 12 months, trades have been made and decisions made that Wayne voiced his opinion against, but they were done anyway," Shumway said. "If Wayne had been making all those decisions, then he should be taking some heat, but the reality is Wayne hasn't had that involvement.
"It's (Gretzky's involvement) grown each year, and this year it'll be the most it's ever been."
There will be a new focus on hockey fundamentals and assessment tests, and prospects will be required to work with a skating coach, said Shumway.
Fired along with Barnett were director of hockey operations Cliff Fletcher, assistant general manager Laurence Gilman and media and player relations director Rich Nairn. Associate coach Barry Smith quit last week.
"I'll be blunt about this," said Shumway. "When I started holding meetings in December, I was stunned that we didn't seem to have a plan.
"There wasn't a vision of what it means to be a Coyote."