Since the current bond buying is $85 billion per month ($45 billion in Treasuries and $40 billion in Mortgage backed debt), they are now buying most of the bonds being issued by government. They are essentially facilitating large government deficit spending; spending that is indeed "showering" money on the economy. This alone is theoretically inflationary in normal times. But the intent of the bond buying is driving interest rates to abnormally low levels.. All of this is known to be inflationary, but we're not seeing high CPI inflation.
But there are early signs of inflation in certain assets. Government spending has continually inflated medical costs, for instance, even in recessionary times. See my blog: Early Warning Signs of Inflation.
Also, the program is causing additional potential problems. By lowering (distorting) interest rates, they have also distorted stock prices (higher), caused underpricing of mortgage rates, etc. Mortgage rates are distorted below profitable levels (taking into consideration default rates) so banks can't compete with the government. Because of this, very few mortgages originate at banks---nearly all come from Fannie, Freddie Mac and FHA--who are absorbing unlimited losses of defaults to the account of the taxpayers.
The Mechanism Of Federal Reserve Money Creation
When the Fed buys bonds, it deposits the money in the accounts of the banks of the securities brokers. Those deposits show up as M1 money, otherwise called "high powered money" or part of the "monetary base." It's called "high powered" because there is a multiplier effect of M1 money. Once this is lent out, it causes about a 4 fold increase in the broader money supply statistics. Since M1 has rocketed up by over $1 Trillion, this means the M2 money could rocket up by $3 to $4 Trillion, which means that the broader money supply (M2) would rise about 50%. This would be highly inflationary. This monetary base been rocketing up since 2008 with Federal Reserve bond buying. Banks could lend this base money out consistent with the reserve requirements of the Federal Reserve, but they haven't for the most part. Because they haven't lent it out is why inflation hasn't taken off.
So, what we have is large amounts of "excess reserves" in our banking system. We almost always have some excess reserves (banks just take what they need to meet the demand for loans) but now it's a very large number. And it's true that it hasn't been lent out to the public and therefore hasn't created a more serious inflation problem. But you could say that it's monetary "tinder" sitting there waiting to be lent out--which could catch fire so to speak. Yes, you could argue that if inflation ticks-up, that the Federal Reserve could just adjust the reserve requirements to slow or halt the loan-making process. But the market perceives an inflation threat down the road. Also some of this money is making it's way into stocks and bonds and causing dangerous asset bubbles. That's why QE can't continue ad infinitum.
Then comes the question of reducing the number of bonds on the Fed's balance sheet. Maybe it can be done without incident, but we're in uncharted waters. We've already seen a huge jump in interest rates (from 1.6% to 2.8% on 10 year treasuries) when the Fed even hinted that the bond buying program might end. Maybe the Fed will just let the bonds just sit there and let the bonds mature. But none of this has ever been attempted on this scale. We're in uncharted territory and PhDs at the Fed are conducting a massive experiment in monetary policy with incomplete understanding of the costs of unwinding of the program or other unintended consequences.
Let's be clear, the PhDs at the Federal Reserve don't really know what they're doing and they are making it up as they go.