The convenience of online shopping hasn’t gone unnoticed by luxury shoppers, 68% of whom say they will share personal data with e-retailers. However, 75% of those shoppers say they share the information only because it’s necessary to complete a transaction. Their hesitations come from privacy and security concerns, according to the Luxury Institute, a research firm.

The Luxury Institute in December conducted an online survey of 1,232 U.S. consumers aged 21 and older with annual household incomes of at least $150,000—representing about the top 9% of U.S. households as measured by income, the institute says—about their preferences for sharing data online versus in stores, and their online tracking preferences. Nearly 60% of the respondents say they believe they have no control over their personal data after sharing it with a company and 30% believe there is a high risk that their personal information will  be compromised.

“Luxury firms must optimize respecting privacy while earning trust in order to collect valuable customer data and use it to create value for customers,” says Milton Pedraza, CEO of the Luxury Institute. “Should privacy legislation be enacted, the brands that will be superbly successful will be those that have built genuine, trusted, long-term human relationships with their customers.”

Only 24% of surveyed consumers say they shared personal contact information, such as an e-mail addresses or phone numbers, while making their most recent in-store purchases, compared with 68% who say they did so during their most recent online purchase. However, among consumers in the survey who report having a relationship with a sales associate, they also say they are more likely to share personal information both in stores (46%) and online (31%).

About one-quarter of luxury shoppers in the survey say they think brands that collect their personal data sell it to another company in order to target them with more personalized advertising. More than half of those shoppers, 59%, say they think they have little or no control over what personal information a brand shares with another company. Just one in 10 shoppers surveyed think brands use the data to follow up on past purchases or assist with future ones, the institute says.

Two-thirds of wealthy shoppers in the survey say they can tell whether a web site is tracking their activities. Of those respondents, 29% say they opt to turn tracking capabilities off—with more men (33%) doing so than women (25%)—21% say they edit the tracking settings and 17% say they chose to leave the tracking abilities fully on, the survey says. (The report did not detail if turning those features off meant disabling cookies or other actions.)

If an online “do not track” list were available, two-thirds of consumers in the survey say they would opt into it; 82% report they are already registered for a “do not call” list to keep telemarketers away. “Given that consumer knowledge about online tracking is quickly increasing, brands and advertisers must proactively prepare and determine innovative ways to learn about consumers and build relationships if online activity can no longer be tracked,” the Luxury Institute recommends.