Now that Uber Technologies Inc. and Lyft Inc. are trading on public markets, the ride-hailing companies are under pressure to achieve years of rapid growth.

First they will have to figure out how to hang onto drivers like Carl Wagoner, a Lexington, Ky., minister and potter who last summer drove for Uber to earn extra cash. The gig proved less lucrative than he thought—about $5 an hour after gasoline, he calculates—given that he avoided the higher-priced, late-night bar crowd. He quit within six weeks.

“Just...