The question says it all - do we get to keep them or do they go away?
At the end of the season hats are collected from all Stack Exchange users, loaded onto special private planes and shipped to war torn parts of the world to give some cheer and warm the heads of the victims of violence. Here are some of the happy children from last year:
No, the hats simply disappear as soon as Winter Bash is over, as explained on the Winter Bash 2016 page:
the hats get put back in their boxes on January 9th!