I'm sure there are loads of ways of dealing with them, but here is my technique.
Using the tag as a template I cut out my paper. This happens to be Turtle Doves from the 12 Days of Christmas collection.
Lining up my cutout with the tag, I press around the grommet hole to slightly emboss the area.
Holding the paper in place I take some sandpaper and lightly sand around the raised area. (Note: I just grabbed what was on hand to demonstrate, this sandpaper is very coarse so it's a bit scratchy).
Using that area as a guide, I cut a hole in my paper.
Now I affix the paper to the tag. I like to use rubber cement or tacky glue. The moisture of these adhesives gives the paper a tiny bit of 'give' which allows the paper to stretch slightly, so I can press the paper firmly into place. I run my nail around the edge to give a nice tight fit around the raised metal grommet.
When the adhesive is dry I run a blade around the edge of the grommet. NOTE: If your paper is wet or damp from the glue it is more likely to tear rather than cut.
Now I carefully cut away the excess paper.
And there you have a perfectly cut grommet hole!
For the back of the tag where the grommet is flatter, I sometimes just sand away the excess paper. Because I'm using very coarse sandpaper for this demo it's really scratchy! If you use a finer grit and some care, the results won't be nearly as messy.
Because I usually ink or grunge up my edges I actually don't mind if the reverse isn't absolutely perfect. A bit of paint or stain will camouflage those edges, and with a ribbon in place no one will even notice ;)
Hope that helps. Happy mess making!