At the heart of part of the debate appears to be a dispute over interpreting results from the Cooperative Congressional Election Studies (CCES), a unique large-scale panel survey of over 20,000 respondents in which (some) self-reports are linked to official voter records.
Several researchers from Old Dominion and George Mason published this 2014 study in which they argue: "The proportion of non- citizens who voted [in the 2008 presidential election] was less than fifteen percent, but significantly greater than zero. Similarly in 2010 we found that more than three percent of non-citizens reported voting."
The PI of the panel study itself and coauthors published a relatively brief rebuttal in 2015, in which they highlight the dangers of scaling up rare frequencies of results that are subject to measurement error. They think (and demonstrate) that "measured" noncitizens who voted just answered the survey question incorrectly. To me, the most convincing look at this was in their Table 2, where none of the people who reported being noncitizens in both the 2010 and 2012 waves of the panel had voted:
Another route to go would be to impute documented status in the survey using other characteristics, such as is done in Census surveys by Jeff Passel and colleagues at Pew and Rob and Bob Warren at the Center for Migration Studies.
