I was in a similar situation, grommet, linen type drapes that I really liked but not solid enough for full privacy. I think part of the draw to them (for me) was that the lighter weight material gave them a less heavy look. If that is something you also like about your drapes, it's something to consider.

As an alternative, I used a double rod and added sheers which ended up being a really versatile option. I can pull them open behind the grommet panels and no one knows they're there, I can use them to filter light, and in the evening they act as a double layer for privacy. I hung them with rings and used a fold at the top so they were just a tad shorter than my grommet drapes (a suggestion by a drapery lady at JCP). It's also a much more cost effective way to get almost the same result for privacy.

I see no issues with adding sheers as noted above, or with adding a liner to your drapes.

If you sew, then I would attach under the grommet rings. If you don't sew, consider double faced Velcro, or iron on fusible hem tape. Or safety pins if you just want to try it out.

Lining should not be attached at the bottom. You want it to hang freely and not skew the line of the drapery.

I sewed insulated panels into my tab top curtains and it was a piece of cake. Tedious, but easy enough- I just used a simple whip stitch- heck, I almost just basted them on to keep them easy to disassemble if I ever needed to.

Just a matter of you deciding what you want :)

Okay, I did one side and it looks GREAT!

I totally cheated and bought 72" wide white felt for $3/yard. I also used sew-on velcro to attach the felt to the tops of the panels. I put a 2" wide piece under each grommet and left the rest alone.

I ran the felt sideways, since 72" is about perfect length for the lining and extends almost 3' below the windows. It works really well, the light is filtered without being blackout. And the curtains feel weightier and more luxurious when they are stacked back toward the ends.

It took 1.5 hours to do the 2 panels on the left ... Now just need to do the 2 on the right. They had the felt in gray, camel, and cream if someone wanted more light blocking than the white.

I will probably tack the sides down by hand, lightly, because I think in the summer I will want the curtains to be unlined (that's part of why I used the Velcro, to make it easy to remove the liners in the summer).