As a volunteer who actually issues petitions to other volunteers, I can tell you that this is not true.
People who sign the petition only need to represent themselves as a registered voter in Senate District 19. We will show them a district map and have them locate their residence on the map if there is any question at all. Unlike an election under the new same-day law, the signature and address of the elector will be validated against the voter rolls by an employee of the Secretary of State. If they are found to not be eligible to sign, the signature won't be counted. In a same day election of course, a same-day registrant elector's vote will be anonymized and counted regardless of their ultimate eligibility. In our case though, we try to screen signers as best we can. Disallowed signatures do no one any good.
We will only ask for your DL if you will be a petition circulator. We have to check before we issue petitions so we can ascertain that you're a state resident and so we can prove that we did so by recording your DL # and DoB on the issuance form. We also train circulators regarding the legalities they must follow when collecting signatures. For instance, they must witness each and every signature. They will sign an affidavit to this effect in front of a notary when the petition is returned.