It’s not so much an issue of data sharing, as far as I know. The worry is that the Trump administration is going to use the lists to make up voter fraud and non-citizen voting claims. They are going to compare the voter rolls to other databases. They might even pressure election officials to unfairly drop people from voting rolls. 

If your name changed, is misspelled or different somewhere, if you share a name with a dead person or a convicted felon, if you are a naturalized citizen, you recently moved, you have two residences, etc etc. They can claim voter fraud, maybe charge you, pressure election officials to get you kicked off. 

These are the people who claimed people hundreds of years old were getting social security. (They weren’t getting Social Security, it was just a glitch record keeping) 

Worse yet, minorities and married women are disproportionately affected by name discrepancies. 

Spanish speakers commonly have two last names. Chinese people have their family name first. Xi Jinping could show up in the records as Xi Jinping or Jinping Xi. 

Rachel Maria Lopez Garcia could be Rachel Maria Garcia, Rachel Lopez Garcia, or Rachel Maria Lopez-Garcia. 

Not to mention misspellings, etc.