I would tend to assume that in each state, the law would require
Your assumption would be incorrect. In many states (probably most U.S. states), a representation of the person registering to vote that they are a U.S. citizen is sufficient to be validly registered to vote, and registered voters are allowed to vote without further verification of their citizenship.
Other information could be called upon to prove citizenship if the eligibility of the voter to vote based upon citizenship is formally disputed.
These seemingly lax requirements are quite effective.
In practice, the number of cases of non-citizens voting or registering to vote is infinitesimal (on the order of one in a million ballots cast or less). Specifically:
votes by non-citizens account for between 0.0003 percent and 0.001 percent of all votes cast [ed. i.e. 1 to 3 per million ballots cast].
The cases of non-citizens voting that do arise mostly involve good faith mistakes lawful permanent residents (i.e. green card holders) who are in the process of becoming naturalized U.S. citizens. In these cases, the non-citizens who try to register to vote or vote misunderstood where they were administratively in the rather complicated red tape of the naturalization process, and become U.S. citizens shortly after registering to vote or voting.
Empirically, there are more conservative activists who vote illegally in federal elections, feeling justified in their cheating as a result of their misperception that voting by ineligible voters is widespread and wanting to counterbalance that, than there are non-citizens who vote illegally in federal election.