Google has a message for New Mexico’s Lea County, an area of arid pastureland in the oil-rich Permian Basin, that the company is eyeing as a possible home for a new data center: “We’re listening.”The technology company reached out to the community via a website, saying its plans are still in the very early stages, but such projects “naturally bring questions about how data centers operate.”It then tried to answer those queries pre-emptively, assuring the approximately 75,000 residents, for example, that it expects its data center would consume water at a similar rate to an office building. A survey asks what additional investment the area might need.

Locals set up a Facebook page within days to organize opposition, and the group now has about 400 followers. With a commission vote on the data center scheduled for November, many residents feel taken aback by how quickly they are expected to grapple with the plans, said Clayburn Griffin, who started the group with neighbors.

Griffin said the call for feedback feels disingenuous. “We’re feeding them information on how to sell their data center to us,” he said.

Americans are growing more concerned about data centers, with more than half thinking they mostly hurt the environment, according to a September report from Pew Research Center.

Hyperscalers, or large tech companies that operate vast networks of computing infrastructure, say they want to be increasingly transparent, getting construction plans out in the open sooner in the hopes of heading off opposition. But answering local queries and frustrations is getting trickier as resistance picks up, with government officials and residents expressing worries about everything from electricity prices to water waste.

A Google spokesperson said its call for public feedback isn’t hollow, and it wants to ensure it can tailor its approach to help the specific needs of the community. The spokesperson said it is following a meeting schedule set by the county.

The promise of a more forthright approach means some public debates are starting much earlier now. Amazon.com, for example, recently said it would no longer use nondisclosure agreements with local officials on its data center plans. Meanwhile, opposition to data centers is picking up ahead of the midterm elections.

“We share information about our plans as early as possible, listen to concerns, and ensure communities have the facts and the ability to engage with us as projects progress,” said Matt Garman, chief executive of Amazon Web Services, in a blog post. Amazon declined to comment further for this piece.

Hitting the right tone on data centers remains an uphill battle as lawmakers increasingly halt approvals: Hundreds of construction moratoriums have been introduced this year. Democratic lawmakers in New Mexico—where Google is eyeing a data center site in Lea County—also say they want to propose a moratorium in the state.

Some companies are scrapping data center plans amid pushback. In the first quarter of this year, local resistance disrupted at least 75 projects worth about $130 billion, according to research firm Data Center Watch.

“There is a fundamental storytelling problem here,” said Stuart McLaughlin, head of advocacy at Google.org, Google’s philanthropic arm, speaking at an event in Manhattan last month. “[We] all need to get way better at telling this story.”Data center operators are holding open houses, inviting residents to air their frustrations—and hoping they come away appeased. They are also offering communities millions in local investments such as free community college and energy efficiency upgrades, and in some cases, pledging to give residents checks if data centers are built.

A Google spokesperson said announcing data center plans earlier does invite more risk, but the alternative would be losing public trust.

Activists opposing data centers say promises to make big community investments and other moves like this show hyperscalers have lost control of the narrative. “Their PR moves have been faceplant after faceplant,” said Mitch Jones, managing director of policy and litigation at nonprofit Food & Water Watch. “They’re not gaining any ground, and they’re growing desperate.”Aside from attempts to calm and entice locals, Amazon took a harder-line approach earlier this month, warning the public that the moratoriums could be harmful.

In a memo, the company asserted that “the U.S. could be writing its own losing ticket” if the construction bans go forward, saying data centers are needed for schools, hospitals and even satellite operations.

Companies are also jostling to demonstrate that their data centers are less resource-intensive than their competitors. Amazon, for example, published a blog post stating its data centers use air rather than water to cool off electrical equipment about 90% of the time.

In Lea County, local commissioner Tyson Pierce, who is also a rancher and owns an oil-field services business, said the talks with Google are going well, but he’s still eager to get more specifics on power and water demands—a particular concern for a community on dry, sunbaked pasturelands.

Pierce said he has also been calling some residents personally to explain why he thinks the deal could benefit the county.

“If you sign this deal and they come in, you get one shot to get it right,” said Pierce. “So make sure you have all your ducks in a row, and get them to do as much as you can upfront.”Write to Clara Hudson at clara.hudson@wsj.com