Our job was to verify who they were. Citizenship and any other qualifications to vote were determined by the voter commission. If you qualified, they would send you a voter ID and assign you to a precinct.
I informed some people they would qualify to vote in Michigan if they were felons and had done their time and paid court costs. In some states, they are not able to vote.
In practice, I don't think it is anybody's responsibility to verify citizenship. In Texas you complete a voter registration application and if you check the "U. S. Citizen" box you normally become a registered voter.
Texas sent in thousands of names to Houston and notified them it contained many non-citizens. Houston challenged this and many of those people were naturalized citizens.