When someone amends the federal return, the IRS routinely notifies the state(s) involved in case they might want to do something about it.
The answer I gave above is still relevant. If the amended return would result in a change in the state's tax due, then you will want to file the amended return for the state sooner rather than later. If you owe the state more money, it will cut off the interest that is running on your tax due, and if you will get a refund from the state, why wait for it - get it sooner than later.
If the amended federal return will not cause a change in the tax due on the state return, then you can wait to see if the state notifies you. If they do, fine, send the state the amended return, but if they don't ever notify you, then it's a win-win - you don't have to file the state amended return, and the state doesn't have to deal with paperwork that won't change anything.
