All transactions are two-sided. However in most cases the market for the object being sold is impacted much more strongly than the market for the object being purchased. If a huge new silver mine started selling silver on the open market, the impact on silver prices would be dramatically larger than the impact on the market for whatever they receive in exchange, unless they engaged in some sort of barter. The same is true of sales of Boeing airliners. Boeing's sales have a big impact on the airplane market, but not much impact on the market for whatever asset is used to buy their jets (say money or bonds.)
From January 2002 to January 2006, the US had a big housing boom (some call it a bubble, I don't.) During this period the Fed steadily increased the money supply, boosting the monetary base by over $9 billion per quarter, a total increase of $149 billion over 16 quarters. And the economy was smaller then, so that would be equivalent to an increase of $12 to $14 billion per quarter today. These injections (OMPs) were used to buy securities---mostly Treasury securities.
To really understand monetary policy you need to think about what makes the Fed unique. Here's another entity that is buying lots of bonds:
The tech giant reported record profits last quarter of $18 billion - the most of any S&P 500 company ever. Apple's blowout quarter single-handedly pushed overall S&P 500 earnings growth into positive territory after the company reported Tuesday night. As of then, without Apple, earnings were projected to contract 0.5%, according to FactSet.
. . .
Also standing at record levels: Apple's cash and marketable securities. That figure ballooned to $178 billion last quarter, up from $155 billion in the prior period.
So Apple is probably buying up even more securities than the Fed did during the housing boom, even as a share of GDP. And of course Apple is hardly unique; lots of other companies buy bonds, and countries such as China and Japan have accumulated far more securities than even Apple.
Because the global debt market is enormous, and because many types of debt instruments are close substitutes, Apple's purchases are not very important. A giant silver mine, or Boeing, or Apple, is important because of what they sell, not what they buy. They have enough market power to influence the price of what they sell, but not enough to significantly influence the value of most securities. (Obviously this would not hold if they accumulated just a few thinly traded stocks.)
And the same is true of the Fed. But wait (you must be thinking) don't Fed OMPs of Treasury debt cause debt prices to rise? Not always, massive debt purchases in the 1960s and 1970s caused bond prices to fall sharply. But yes, in many cases it is true that Fed OMPs do cause bond prices to rise. However that's not mainly because of what the Fed is buying, it's because of what they are selling. Even when central banks used to buy gold instead of Treasury bonds, interest rates tended to fall immediately after monetary injections, due to the liquidity effect. Until prices adjust, interest rates must fall until the public is willing to hold the larger cash balances. So the direct interest rate effect of Fed bond purchases is trivial, just as the interest rate effect of Apple's bond purchases is trivial. It's the money injections that matter, and interest rates may rise or fall on those injections, depending on whether the liquidity or Fisher effects dominate.
Does this change at the zero bound? Perhaps, but not necessarily in the way that you'd assume. Over the past 6 years the Fed bought far more debt than did the ECB, and that's why longer-term risk free interest rates are higher in in the US than the eurozone. The Fed policy led to faster NGDP growth in the US than the eurozone.
Central banks are unique because of what they sell---they have a monopoly on base money production---not because of what they buy. There's a reason it's called "monetary policy" and not credit policy.
PS. It's possible that the huge Potosi silver mine depressed the value of money in the 1500s, but that's because in those days silver was money. It's the supply and demand for the medium of account that matters; what they buy with the new money is of trivial importance (unless it's something with a much more thinly traded market than bonds.)
PPS. When the financial press says "cash and securities" I believe the term 'cash' generally refers to short-term debt such as T-bills, not actual currency. Someone please correct me if I am wrong about Apple's "cash hoard".