In Denver, the partnership is centered on ticketing rather than car rides. Through Uber’s app, people get a new way to buy tickets and obtain train and bus schedule information. Uber doesn’t make money selling the tickets, but it benefits when ticket buyers, like Ms. Ellis, stay in the app to book a ride from the train station to their destination.

One of Uber’s earliest partnerships was in 2015 with Dallas Area Rapid Transit. That year, DART agreed to temporarily display Uber rides as an option in its app during St. Patrick’s Day festivities. The promotion, intended to give boozy celebrators more choices for getting home safely, became so popular that DART eventually integrated Uber into its app permanently.

DART now subsidizes shared Uber rides within a few miles of several public train stations. The agency estimated that it spent $15 per rider when it ran bus routes in those areas; now it saves money by paying Uber less than $5 a person.

Dallas transit officials were initially cautious about the partnership, they said. “For a while they ignored us. Then they cannibalized us. Now they want to work with us,” Todd Plesko, DART’s vice president of service planning and scheduling, said of Uber and Lyft. “It’s the kind of market for trips they never did before.”

Uber was also hesitant to share data about riders and routes, citing privacy concerns. Mr. Plesko said Uber had mentioned the hunt for Osama bin Laden as an example of how individuals could be identified from their data. (Uber said no one had used a Bin Laden reference.)

But Dallas ultimately decided to work with Uber. Integrating Uber rides into DART’s app could help stem the flow of riders who abandoned public transportation for private ride-hailing services, Mr. Plesko said.

“If we’re going to survive as an agency, we have to be willing to innovate and take risks,” he said.