Nintendo going after ROM sites is possible, they've been doing that for a while, but regular users getting arrested because they've downloaded something from one of these sites is grade A fearmongering.
First of all, I'd be surprised if these sites kept any kind off detailed logs in the first place. Second, even if Nintendo could get its hands on internet connection records, and that's a big if, they could likely only show that you visited a particular site, but not whether you downloaded a particular file. Visiting ROM sites isn't illegal. Third, even if none of this is true, it will then depend on whether this conduct is a criminal offence in the user's jurisdiction, which it very well may not be.
Overall I think Nintendo will have a very hard time showing that a particular individual violated their copyright.