He met with reporters at the Dolphins' training facility in Davie, Fla., where he said the experience of being moved felt like "being drafted again."
Rosen, 22, was the No. 10 overall pick in the 2018 NFL Draft. He started 13 games during his rookie campaign for the Cardinals, posting a 3-10 record while throwing 11 touchdowns and 14 interceptions.
The Cardinals went with a quarterback once again this off-season, selecting Oklahoma's Kyler Murray with the No. 1 overall pick in the 2019 NFL Draft. Arizona's selection of the Heisman Trophy winner spelled the end of Rosen's time in the desert.
Rosen watched it all unfold while surrounded by his friends Thursday at his home in Scottsdale, Ariz.
"Actually, it felt like I got drafted twice," Rosen said. "They actually called me from their draft room, so I talked to the general manager, head coach and coordinator, just like if I was actually normally drafted."
Miami headed into the 2019 NFL Draft with veteran Ryan Fitzpatrick penciled in as the starting quarterback on their depth chart. The Dolphins signed Fitzpatrick to a two-year, $11 million contract in March.
Rosen said that he thinks he is the Dolphins' backup quarterback "on paper," but first-year head coach Brian Flores says the team doesn't have any starters at this point in the off-season.
"I think that's healthy to have good competition everywhere," Rosen said. "Whether I was the first or seventh string, I wouldn't change what I did every day."
Flores called Rosen a "talented player" with "a lot of arm talent." He also lauded Rosen's leadership qualities and said he is a good addition for the Dolphins.
"Josh will come in and compete," Flores said. "There are no starters. The guys who produce on the practice field and do all of the things that will help this team win, those are the guys who are going to play."
Rosen also completed 55.2 percent of his throws for 2,278 yards during his rookie campaign. He said stepping in behind Fitzpatrick is similar to what he had to do with Sam Bradford in Arizona and he plans to approach the situation in the same manner.
"I think [you have] got to handle what you can handle," Rosen said. "I think it's not going to change anything football-wise what I do. I'm going to try to complete every ball and do every play really well and do what the coaches ask of me.
"Obviously, you have to be a little bit conscious of the leadership aspect. You can't be breaking down team huddles as a backup. There's definitely a sense of awareness that you've got to understand where you position is on the team. But I'll always, regardless of that, try to sort of inspire and motivate and push everyone in a positive direction."
Rosen said he still believes he can be a franchise-type quarterback for the Dolphins. General manager Chris Grier said the team didn't go out saying Rosen had to be a franchise quarterback when they acquired him.
"I would say for us, we looked at it as an opportunity to add a good, young football player that has a lot of potential in this league," Grier said. "We didn't go out saying he has to be a franchise quarterback for us.
"For us, it was he's a very talented young player, still has a lot of upside in the league and the terms for us, the value and taking on the contract and etc. For us, the value was tremendous that we couldn't afford to pass up."
The Dolphins sent the Cardinals a second round pick in the 2019 NFL Draft and a fifth round pick in the 2020 Draft to acquire Rosen. Arizona used the second round pick to select University of Massachusetts wide receiver Andy Isabella.
Miami begins OTA off-season workouts May 13 and has mandatory minicamp from June 4 to 6 in Davie.