Let's start out very simple.
The central bank issues banknotes, and those banknotes are the only type of money that people use, to buy and sell everything else. It really doesn't matter if people keep those banknotes in their pockets or if they keep them in a shoebox at the central bank with their name on it. If they keep them in their pockets, they buy things by putting their banknotes in the seller's pocket. If they keep them in a shoebox at the bank, they buy things by sending a form letter ("cheque") or email (Interac) telling the bank to take the notes out of the buyer's shoebox and put them in the seller's shoebox. Pockets and shoeboxes are all the same. And it really doesn't matter if the shoeboxes and notes all get burned in a fire, as long as the bank keeps a record of how much virtual money is in each person's virtual shoebox.
Remember that MV=PT. Stock of Money times Velocity of circulation = Price level times Transactions.
There are two ways the central bank can increase MV:
It can increase M (by printing more money and buying something with it). Each individual wants to get rid of that extra money, and can get rid of the extra money by spending it, though this is impossible in aggregate, though their attempts to get rid of it by spending it cause PT to rise. We call this the "hot potato" effect.
It can increase V. There are two ways it can increase V.
One way to increase V is to tell people it will increase M in future, so that people expect P to rise in future, so the increased expected rate of inflation reduces the real rate of interest people earn by holding money in their pockets or shoeboxes.
The second way to increase V is to reduce the rate of interest it pays people for holding money in their pockets or shoeboxes. It could reduce that rate of interest below 0% if it wishes. Silvo Gesell advocated paying a negative interest rate on holding money, precisely to increase V.
Again, each individual can get rid of unwanted money by spending it, though this is impossible in aggregate, though their attempts to get rid of it by spending it cause PT to rise. Again, we call this the "hot potato" effect.
If people hold money in their shoeboxes it is administratively very easy for the central bank to pay positive or negative interest on the money people hold. Computers can do this easily. But if people hold money in their pockets it's administratively harder to pay positive or negative interest.
Now let's complicate the story a little.
The central bank decides to retain a monopoly on the money-in-pockets business, but decides to privatise the money-in-shoeboxes business. (Yes, this is totally ahistorical; the commercial banks existed before the central bank, and used to issue their own money-in-pockets.) Several competing commercial banks take over the money-in-shoebox business, issuing their own (virtual) notes that are only allowed to be kept in shoeboxes at the issuing bank. But the central bank makes one exception to its decision to get out of the money-in-shoebox business; it will allow the commercial banks to keep the central bank money they own in a shoebox at the central bank, if they want. Regular people must keep their central bank money in their pockets.
Each commercial bank promises to fix the exchange rate between its own money and central bank money at one-to-one. It does this by promising to swap the two monies in either direction. The central bank makes no such promise (though it promises to help the commercial banks keep their promises in an emergency, by acting as lender of last resort, provided they get their act together quickly). This asymmetrical hub and spoke system of fixed exchange rates, where it is the commercial banks and not the central bank that is responsible for fixing the exchange rate, is what gives the central bank its power. This is what makes the central bank the alpha leader and the commercial banks the beta followers. The alpha central bank can make its money more valuable or less valuable; the beta commercial banks just have to follow along, to keep their exchange rates fixed.
Regular people can use either commercial bank money-in-shoeboxes or central bank money-in-pockets to buy things from each other. One is more convenient in some cases, and the other is more convenient in other cases. The two types of money are substitutes, but not perfect substitutes. The commercial banks are in direct competition with each other, but in limited competition with the central bank, for regular people's money business.
But the central bank is the bankers' bank. The commercial banks use central bank money between themselves. If a regular person buys something using BMO money-in shoebox from another regular person who wants TD money-in-shoebox, the two commercial banks transfer central bank money from the BMO shoebox to the TD shoebox at the central bank.
Commercial banks can pay any interest rate they want, positive or negative, on their money-in-shoeboxes, subject to direct competition from other commercial banks, and subject to limited competition from the central bank. (And they can charge any other fees they want, like fees for sending them form letters or emails.) The central bank can also pay any rate of interest it wants, positive or negative, on its money-in-shoeboxes. (And shoeboxes can contain negative amounts of money, or red notes with negative value, called "overdrafts", with a spread between the interest rate on positive balances and the interest rate on negative balances.) But it is administratively difficult for the central bank to pay interest, positive or negative, on money-in-pockets. So that money always pays 0% interest.
Commercial banks, just like the central bank, can increase their M by printing money and buying something with it. And what commercial banks normally buy, and what the central bank normally buys, is non-money IOUs. But the central bank normally buys IOU's signed by the federal government; and commercial banks normally buy IOU's signed by regular people. They call this "making a loan".
Which is basically what the Canadian monetary system looks like.
If there were no money-in-pockets, so regular people only used commercial bank money-in-shoeboxes, and commercial banks only used central bank money-in-shoeboxes, it would all be quite simple. If the central bank wanted to increase MV, it could increase M by buying something now, or increase V by promising to increase M in future, or increase V by reducing the interest rate it pays on money. This creates a hot potato effect between the commercial banks, as each commercial bank tries to get rid of central bank money. But that creates a second hot potato effect between regular people, because the way commercial banks try to get rid of their central bank money is by buying IOUs and by reducing the interest rates they pay on their money.
Hot potato central bank money creates hot potato commercial bank money. Which is exactly what the central bank wants, if it wants to increase PT.
And it makes absolutely no difference to this story whether nominal interest rates on holding money are positive or negative.
But the central bank money-in-pockets messes up this nice neat pyramid of stacked hot potatoes. Because the nominal interest rate on money-in-pockets is stuck at 0%. So if interest rates on money-in-shoeboxes fall, while the interest rate on money-in-pockets stays the same, commercial banks will face greater competition from central bank money for regular people's money business. Which they won't like.
If the two types of money became perfect substitutes at a 0% rate of interest, so that regular people would switch completely to using money-in-pockets if money-in-shoeboxes paid a negative rate, this would be the end of commercial banks. But they very probably aren't. We haven't seen much sign of people doing this as nominal rates of interest have fallen a lot over the last few decades. And money-in-pockets just seems so inconvenient for many purposes.
But there may be a psychological barrier from banks' customers to being paid negative rates of interest. A bit like a mirror-image of the old prejudice against usury on "barren" money.
And reducing the interest rate paid on money is only one way for central banks to create a hot potato and increase MV.
[This is my version covering the same ground as Scott Sumner and Frances Coppola. (I love how Frances talks about banks playing "pass the parcel", but I must insist on "hot potato" as the traditional metaphor, and in pass the parcel you want to be the one holding the parcel when the music stops, while in hot potato you want to avoid holding it).]