No, studies have been done, and routinely show that non-citizen voting in election accounts for a pittance of the vote (e.g., "A study of 2016 data by Brennan Center for Justice from 42 jurisdictions found an estimated 30 incidents of suspected noncitizen voting out of 23.5 million votes cast (or .0001% of votes)"); historical data of prosecution of non-citizen voters agrees with the studies, similarly showing it to be non-issue.
There is evidence, the "one party" routinely presents it. This isn't some "both sides" issue.
> the other party […thinks] the solutions to proving legal voter status would be difficult for their citizen voters to accomplish
Yes, because they would. The GOP's proposed "SAVE Act", which was pushed directly by Trump, for example, would have (effectively) required most citizens to present passports as proof of citizenship. (And further, to obtain a passport, since most Americans do not have one.) (Among other problems with the bill.)
(And just to note, since there is some non-US in the thread: you have to be a citizen to register to vote, and — though this is handled on a state by state basis, but I do not think any state deviates here — you must prove that already at registration. The ongoing debate above simply makes that harder to do.)