I'm going to have to file a 1099 for someone that I'm paying money to this year so I was just reading the instructions, and they say that I am supposed to send one copy to the taxpayer, one copy to the IRS, and one copy to the state tax agency. It may be that some states require it and others don't, I haven't investigated that.
I see some of the commenters assume that you're asking this because you want to not report your 1099 income to the state. If this is what you're thinking, I can only say: Don't. Even if the state doesn't get a copy of your 1099, they communicate with the feds. The information is available to them. They'll track you down sooner or later and then you'll be facing penalties and interest and possible criminal prosecution.
If that's not what you were thinking at all, my apologies for impugning your motives. :-)