At the Rock County Democratic Party Headquarters in Janesville, Wis., Wanda Sonnentag calls voters encouraging them to vote. (Scott Olson/GETTY IMAGES)

Dozens of men, women, children and dogs showed up early Monday morning outside the Madison Labor Temple, where labor-backed organizers sent them out in search of Democratic votes. Their goal was written in chalk on the sidewalk at their feet: “Barrett or Bust.”

If Milwaukee Mayor Tom Barrett (D) can pull off a come-from-behind win in an election to recall Gov. Scott Walker (R) on Tuesday, it will almost certainly be because of volunteers like these, whom Democrats are counting on to overcome being outspent by tens of millions.

The long-awaited vote is viewed here as a referendum on Walker’s move to curtail public workers’ collective-bargaining rights and a harbinger of whether Republicans have a shot at winning Wisconsin this fall for the first time since 1984.

But the contest also will be an early test of a dynamic that both parties expect to play out in a dozen or so battleground states in November: the effectiveness of the Democrats’ ground organization against the expected advantage Republicans will have in fundraising and on TV.

The importance of on-the-ground organizing for President Obama’s reelection effort was made clear Monday in a Web video released to supporters.

“You know what really matters in a really close election? The unprecedented grass-roots organizing we’re doing every day in states across the country,” campaign manager Jim Messina said at the outset of a three-minute video in which Wisconsin was listed as a toss-up state. “We’re following the strategy we’ve had from day one, and we can’t afford to lose focus on that.”

With Walker holding a more than 7-to-1 advantage in the money race — and with GOP-aligned outside groups far outspending their counterparts — Democrats maintain that their chance at victory depends on a superior get-out-the-vote operation.

Kristen Crowell, executive director of We Are Wisconsin, said in an interview Monday that the labor-backed group is on target to knock on 1.4 million doors and make 1.5 million phone calls. She said the group has about 50,000 volunteers and has spent $2.8 million on its field operation in the past month.

“From day one, there’s been a strong, strong commitment to fund and prioritize the field campaign,” she said.

Over the weekend, Barrett’s camp dispatched more than 10,000 volunteers, who knocked on about 948,000 doors and made nearly 890,000 phone calls. Spokesman Phillip Walzak said those numbers are expected to “more than double” by the time polls close Tuesday night.

Wisconsin officials are predicting a voter turnout of 60 percent to 65 percent — more than in the 2010 midterm election, when turnout was about 50 percent, but less than the 69 percent during the 2008 presidential vote.

According to the nonpartisan Center for Public Integrity, candidates and independent groups have spent more than $63.5 million on the recall effort, making it the state’s most expensive election ever.

A little more than half of that amount — about $34.5 million — has been contributions to the candidates, with Walker taking $30.5 million compared with Barrett’s $4 million.

The remaining $30 million or so in expenditures has been made by outside organizations, according to the Wisconsin Democracy Campaign, an independent tracking group. And the bulk of that $30 million has been spent by GOP-aligned organizations on behalf of Walker.

Republicans are by no means ceding the ground game to Democrats — and they argue that if Walker wins, it will be the result of a top-notch effort by their party on the ground and on the airwaves.

Republicans have opened more than 20 offices across the state. And Wisconsin Republicans, together with the Republican National Committee, have made more than 4 million voter contacts — twice as many as they made before the 2010 election, and a figure that rivals Democratic efforts this year.

“We have the largest grass-roots organization Republicans have ever had in Wisconsin,” state GOP Chairman Ben Sparks said.

Barry Burden, a professor of political science at the University of Wisconsin at Madison, said that the GOP’s overwhelming advantage on the airwaves has forced Democrats to rely more on their field organization.

But he also noted that the conventional wisdom regarding Democrats’ ground-game advantage has come into question.

“Democrats tend to do things more publicly with open-air rallies and volunteers canvassing through neighborhoods,” he said. “Republicans have been using phone banks for months. Their operation is less public but has resulted in over 4 million calls over the past several months. Their efforts seem to be working. Walker got about as many votes in his uneventful primary as the Democrats did in their more seriously contested race. Democrats have been late to the game when it comes to to turnout. “

A Walker victory would most likely lead Democrats to make the case that a fight they had picked — and until recent weeks had expected to win — was lost in large part because of a flood of outside spending.

A Barrett win could lead Republicans and GOP-aligned national groups to further make the case against the powerful labor organizations that they argue were the reason Walker pursued his collective-bargaining reforms.

In the meantime, with independent polling showing Walker ahead, Republicans say they are cautiously optimistic.

Crowell, the We Are Wisconsin executive director, said that those recent polling numbers have only energized volunteers.

“We feel very confident and very good about the position we’re in today,” she said. “We’ve got a fabulous plan. We’re going to beat expectations.”

Sonmez reported from Washington.