I was recently convicted of shopllifting. My question is do I have the right to keep my name out of the newspaper in the local section?

You should have requested this BEFORE the judgement, bit late now.

No. This is public information and the public is entitled to know just who was convicted of what. Not all communities do this sort of thing and I wish more would do this. The Sheriff in my county is now posting warrants with actual photos of the people on his Facebook page. It's amazing how many wanted folks are turned in by their "friends and neighbors".