The first way is knit to the exact column where you see the error and purposely drop stitches until you get to the mistake. That's the easy part - the hard part is making sure you pick up THE RIGHT FLOAT as you are picking up the dropped yarns again. For instance if your mistake is 4 rows down and you're knitting with 2 colors per row there will be 8 strands of yarn and you'll only use 4 of them when picking up the stitches again. I'll get photos of this process soon - it really is a challenge to get it right.
I use basic duplicate stitch to fix colorwork mistakes that I notice long after the possibility of dropping stitches is over. That's what happened with the Christmas Bells mittens - I actually didn't even notice the mistakes until I looked at a photograph of the mittens.
So here I goofed and knit a "U" instead of an "O".
I never anchor the yarn by weaving in the ends before I start duplicate stitching - instead I wait until after I'm done to secure both ends. Usually to start I just thread the needle and insert it from the outside several inches from the spot I'm fixing. That way I can use both ends to adjust the tension after I'm done with the duplicate stitch so it looks its best.
The yarn starts from under the stitch to be fixed and goes around just like the original knit stitch. I always do duplicate stitch from right to left, from bottom to top. It looks much neater when you always do it in the same direction.
Fixed.