On the main Winter Bash page, I can easily see which hats I've earned on at least one site.
How can I find out which hats I've earned on a given site? I'd have expected this to be visible in the hat popup on my profile, but all it gives me is the lists, and I need to click on each hat to see whether I've earned it on that particular site.
You can see your own Top 9 hats on the site in question's leaderboard. (Get to any leaderboard by going to http://winterbash2015.stackexchange.com/leaderboard/[SiteURL], or just by going to the main leaderboard and clicking the one you like best.)
That's about all we've got for now. It'd be a [feature-request] to get something more comprehensive (and you're welcome to request it).
This information exists on the hat rack, but it's not displayed in any way. There are already two different kinds of highlights (the hat you're wearing and the hat you've selected); adding a third would just be confusing. Even more so, any way of showing it means extra stuff on an already busy interface, and if we're honest, this information is mostly interesting for users who are a bit nostalgic for the 2012 Winter Bash rules.
So if someone feels like playing with user styles, look for the CSS class wb-earned-here on the hat rack. I added that class in the hope that we'd come up with a non-obtrusive way of showing this, but we haven't.
