The answer to this question depends largely upon which ridesharing service you are using. For the two market leaders, the coverage they provide for their drivers is largely the same. For both Uber and Lyft, the company-provided insurance covers injuries suffered in accidents for which their driver is liable so long as the app is active and the driver is either on the way to pick up a passenger or the passenger is already in the car.
The only part that matters to you as a rideshare passenger is that once you are in the rideshare car, Uber or Lyft provide liability coverage for your injuries of up to $1 million per incident. Uber also provides $1 million per incident in uninsured/underinsured motorist liability coverage if the Uber car you are riding in is involved in an accident where an uninsured or underinsured motorist is at fault. Lyft provides the same coverage. For both companies, liability coverage is different in New York City because of local laws and regulations.