This is (primarily) for Brits.
1. Suppose I normally buy 10 apples at $1 each. Then the government taxes me $3, buys 3 apples, and gives me those 3 apples. I will now buy 7 apples instead of 10. The net result on my consumption of apples, and everything else, is zero. The only difference is that the government is doing part of my shopping for me.
Fiscal policy won't work in that example, because I offset it 100%. But this post is not about that example; I only used it to help illustrate my second example:
2. Suppose the government of Canada normally buys 10 new Canadian bridges at $1 billion each. Then the Bank of Canada prints $3 billion, and instead of lending it to the government (buying government bonds) like it normally does, it buys 3 new Canadian bridges. The government will now buy 7 new bridges instead of 10. The net result on building of bridges, and everything else, is zero. The only difference is that the Bank of Canada is doing part of the government's shopping for it. [Update for clarification. Printing the $3B will have an effect, but the Bank of Canada buying bridges instead of buying bonds will have no effect.]
Remember that the government of Canada owns the Bank of Canada, so that any revenue the Bank of Canada earns by printing money and buying bridges or bonds ultimately ends up in the government's pocket anyway. So it's exactly like my first example, in terms of who ultimately pays for the apples or bridges.
If you expand the central bank's power to include buying bridges, we should expect to see 100% fiscal offset for any exercise by the central bank of its new power.
My first example isn't very realistic. The sort of things the government buys for us are usually different from the sorts of things that we buy for ourselves. (That doesn't necessarily mean they are less useful, like Christmas gifts where we would have preferred the cash, because some goods are non-rival or non-excludable, which creates micro-public-financy problems if we buy them ourselves.) They won't necessarily be close substitutes, and might even be complements.
My second example is much more realistic. Or it would be realistic if the Bank of Canada actually did buy real newly-produced investment goods. Both the Bank of Canada and government of Canada would presumably choose whether to buy bridges or sewers using much the same sort of criteria. (Though there is that democracy thing, which could perhaps be argued either way.)
If you think the government should build more bridges, then just say so. And if you think you are going to be the next PM, then just say you are going to do it when you are PM. It's really silly to say that when you are PM you will order your central bank to do your shopping for bridges for you. Don't you have a Ministry of Works (or whatever they call it nowadays) that specialises in buying bridges? Because the monetary economists who work at central banks don't normally take courses in buying bridges. And if you are not PM, then it wouldn't work anyhow, because of fiscal offset.
I do regularly read Simon Wren-Lewis' excellent blog, as well as the Daily Telegraph (and sometimes the Manchester Guardian), but I can't claim to be up to date on all the twists and turns of this story. But "silly QE" sounds to me like a more sensible name.
[I was tempted to make a reference to Becker's "Rotten Kid Theorem", which I think is relevant, but was afraid people might take it the wrong way.]
Update: Here are two recent posts that are also relevant:
1. "Interest-free loans from the central bank to the government". (That post was supposed to be about an obscure bunch of Canadian monetary cranks, but it applies equally well to "People's QE".)