John McCain and Marco Rubio weren’t going to follow President Barack Obama’s lead.

When the White House announced that Obama would open a campaign for immigration reform with an event in Las Vegas Tuesday, the Republican senators and their bipartisan working group decided to rush out their plan ahead of him on Monday, according to sources familiar with the effort.

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This game of leap frog was a preview of the personal politics that undergird the push this year to overhaul the immigration system.

(Also on POLITICO: Pitfalls that could stop immigration deal)

Whether he likes it or not, the president’s top legislative priority rests in the hands of McCain, his former 2008 rival, and Rubio, one of GOP’s leading candidates to take back the White House in 2016. That means the odds of passing a bill depend on whether the key players can not only resolve major policy differences, but navigate the tricky dynamics among them.

The long-term stakes are enormous. For Obama, it’s about cementing his party’s lock on the Hispanic vote and finally making good on an unfilled campaign promise from his first run. For McCain and Rubio, it’s about redeeming their party with one of the country’s fastest-growing voting blocs whose alienation threatens to freeze Republicans out of the White House for years to come.

(PHOTOS: Pols react on immigration)

Both parties want Latino voters to give them the credit for solving the problem — or at the very least, absolve them of the blame if nothing comes to pass. At the same time, Obama, McCain and Rubio each face a crucial calculation of their own — how much jockeying they’re going to do and how much credit they’ll be willing to share across their personal and political divides to get the deal they all say they want done.

There was a lot of talk Monday of bipartisan momentum, as McCain and Rubio joined Democratic Sens. Chuck Schumer, Dick Durbin and Robert Menendez in announcing a set of principles on reforming immigration, and the White House welcomed it as a helpful development.

But in reality, each side — Republicans and Democrats, Obama and Congress — is watching the other warily, judging motives and preempting moves in hopes of gaining the upper hand. The Senate group jumped ahead of the president, one Senate Republican aide involved in the process said, because their proposal would’ve been “lost in the noise.”

(Also on POLITICO: Senators strike positive note on deal)

“If you wait, it is like an afterthought,” the aide said, describing it as a group decision.

At the center are two people whom the White House really doesn’t trust, but whose absence would raise serious questions about the viability of immigration reform.

McCain is the lead Republican negotiator and old guard warrior, returning to the fold after years of keeping his distance from immigration advocates in an attempt to seal his legacy. He’s been a constant source of irritation to the administration, most recently for his role in derailing U.S. Ambassador Susan Rice’s bid to become secretary of state — which the president took personally.

Rubio is the fresh-faced ambassador to conservatives whose involvement will signal whether the bill stands any shot of winning broad Republican support. Obama, in one stroke last summer, undercut a legislative effort by Rubio to halt the deportations of younger undocumented immigrants by announcing that administration would act unilaterally on it.