The Fed doesn't "give" money to anyone. They buy treasury bonds from the US government.
The Fed conducts open market operations, and buys treasury bonds on the open market, directly. Because the goal is to keep the sovereign bond yields low. (To steer investors toward riskier investments with higher returns/yields.) So they basically buy up assets (return producing things) for cash (which loses value with inflation).
They then sit on the assets. (People say it's on their balance sheet, and that's why it's sort of an expansionary monetary policy, because it expands the central bank's balance sheet.) That's why it's not free money to the Government. (Whereas in Venezuela, the Government directly prints money for itself to pay wages and utility bills. Basically seigniorage.)
Assuming QE ever ends of course.
The purchased assets (treasury bonds and so on) are idling on the Fed's balance sheet. Eventually as bonds mature the Fed just cancels the newly-created-debt it used to buy the bond, and the asset just disappears from the balance sheet.
The important thing to notice is that this is all a marginal effect. The Fed bought stuff already on the market, so the asset holders already wanted to liquidate those assets, but with the Fed entering the market and driving [bond] prices up (thus keeping [bond] yields down) investors can't just roll over the money, and other investors are discouraged to invest into low-yield assets, thus they try to invest into other stuff, bonds of riskier companies, foreign markets! (The QE programs drove yields down all across the globe, foreign currencies up http://www.bloomberg.com/news/articles/2015-01-23/rupiah-phi... and as a QE program ended this caused problems http://www.reuters.com/article/2013/08/19/us-india-markets-r..., because governments - just like in pre-2008 - saw cheap loans and went to town.)
So, all in all, money goes where the investor things it'll grow best (low righ and high yield). Real estate was that after 2000, nowadays it's all over the place. (RE is picking up again, look at the insane prices in SF and London; VCs pour billions into emerging sectors - look at the valuations here on HN; and naturally people look for riskier things, abroad, which is good for the global economy.)