The first incident involved an Uber driver getting arrested and charged with the rape of a teenage girl.
In the second incident, which occurred on Sunday, an Uber driver was charged with lewd and lascivious behavior. A teenage girl reported him to police, saying he exposed his genitals to her. The Everett Police Department identified the man as Paul Griffin, 28, of Malden, and said he was due to be arraigned on Monday.
The first driver had an "extensive" criminal record, while the second driver had "several open cases" similar to the charges he was arrested on, DeMaria said, calling both incidents "extremely disturbing."
The drivers "never should been allowed the opportunity to drive for your company," DeMaria wrote.
"It is clear to me that the current driver screening system is simply not working, and recent ride-share legislation passed here in Massachusetts will not be implemented for several months," he said in his letter to the Uber CEO.
DeMaria wrote he is interested in establishing a local licensing process for Uber Drivers and to "partner with you to better protect our community."
An Uber spokesman didn't immediately have a response to DeMaria's letter.
According to Everett Police, patrol officers attempted to stop the second Uber driver, but the vehicle accelerated into oncoming traffic, turning down a dead end before the suspect fled on foot. The officers eventually caught up with him and arrested him, police said.
According to Suffolk County District Attorney Dan Conley's office, Griffin has an open Boston Municipal Court case for "open and gross lewdness." He was arraigned on March 26 for allegedly exposing himself to two women in downtown Boston, and bail was set at $500.
DeMaria in a statement said the state Department of Public Utilities has not yet implemented new regulations requiring criminal background checks for drivers with ride-hailing companies like Uber and Lyft.
The mayor again called on the state Legislature to tweak the law and include fingerprinting as part of the background check.
Boston Police Commissioner Bill Evans has also criticized the law over the lack of a fingerprinting requirement.
Gov. Charlie Baker signed legislation in July setting up a two-tiered background check system. The company would complete one and the state Department of Public Utilities would also complete one on drivers.