I am noticing that the pen blank on the left side nearest the head turns smooth but the pen blank on the right next to the set screw vibrates even after the blank is round. Why is that? Is there something I can do?
Make sure your alignment is also checked. There is a nice "how-to" on Russ Fairfields site about how to check your mandrel alignment, to see if your mandrel is bowed. I did it, and it made all the difference in the world. And as stated above, tailstock too loose or too tight can cause similiar problems.
Ditto above. How do you square your tubes? If you don't use a pen mill you may have ends that are not 90 degrees. When you tighten the nut on the mandrel you may introduce a bow in the mandrel only during turning because you don't have exact 90 degree angles on every end piece.
What I do to check my mandrel is take a Sharpie marker and touch the mandrel while it is spinning (with no wood). I rest the marker on the lathe tool rest. I do this is several spots. Then, I turn the lathe off and check the line on the mandrel for uniformity. When the line is good, so are my pens.
Mike