I don't know if you knew this but lottery money is essentially public money so your name is, quite literally, plastered everywhere so people can know who you are. You won't have that provacy, unless you move to a different country or something
I think you'd have to go out of your way to find out whether or not someone won the lottery. For example. I don't live in florida but I know a shyt-ton of people win the lottery from that state. As long as its not some "record breaking" lottery amount, word won't spread that far unless you're deeply invested in the "goings ons" of the lottery community.
For example. I don't live in the same state that my parents do. In this state there are some lottery games that are only played here. Even if someone from my home state of California followed the lottery closely, why would they care to check if someone won some lottery that is only available in a handfull of states and not their own.
Currently the powerball is at $120million. Thats a hell of a lot of money, but its not a record breaking amount by any stretch of the imagination. If someone that follows or plays the lottery found out that someone won that lottery, (and they live in another state) they'd probably just mumble
"fukk" under their breath and move on. Also, its very rare for someone to instantly go and pick up their check so in the two weeks or month or so that is takes for that person to pick up their check everyone would have forgotten that they even won.
Another thing you
could do (if you were extremely paranoid) is hire a lawyer to pick up the check and have them cash it under the name of a business so the name of your company (fluff-ball incorporated or whatever) is used instead of your real name. So, yeah i'd say its pretty easy to formulate an "escape plan" to avoid people once the discovery of your winnings are made public.