Good question, with all the foreign cheese on shelves of US stores, of course the reciprocal question is how much exported from US which of course is harder to see.
I found UN database http://data.un.org/Data.aspx?d=ComTrade&f=_l1Code:5
I used left side filters Country = USA, Year = latest data is 2006, and More filter on 5 categories of cheese that I could see: Fresh, Grated, Processed, Blue Veined and basically Everything Else, the by far the largest category. Sadly not the brie vs cheddar that I think you are looking for. For "Everything Else" for 2006 cheese imports to US were 183,983,465 kg (404,763,623 lb), re-exports were 1,019,101 kg (2,242,022 lb) and exports were sadly only 32,438,447 kg (71,364,583 lb). This means slightly over 1 lb of imported cheese for every American resident, which seems low as a % as we eat way more than 1 lb per year in our family
. Do you have any info on US cheese consumption? Anyway, in summary, for this category in 2006, the US imported (after re-exports) 5.65 times as much as exported! A bad example of the US's poor deficit of trade . . .
But! It's getting better, here is an article on 2007 "Exports increased 24 percent by volume and 59 percent by value over 2006" http://www.flex-news-food.com/pages/15317/Dairy/Milk/USA/us-dairy-exports-grow-2007-likely-2008.html
, it's a start, but a long road up, maybe the article's cited report has more detailed info than the UN's website?
Hope this helps . . .