So is it the case that at any time, anyone can move any amount of BTC that they control into any sidechain?
Yes, and in doing so they give up access to those bitcoins. So the amount of total bitcoin-equivalent coins under control by users is conserved.
EDIT: What I am getting at is whether there are limits to the number of units on a particular sidechain.
Well there obviously can't be more than 21 million * peg price sidecoins. Other than that, the number of sidecoins in existence is simply determined by current demand. It helps to think of sidecoins and bitcoins being the same thing, as distinctions between the two are mostly implementation details. At any given time there are max 21 million bitcoins in existence, some of them on the main bitcoin chain, some of them on specific side chains.
One does not need to find a "seller" or "redeemer" of sidecoins to make a BTC-sidecoin trade, since the sidecoin is not truly a separate unit of currency, so one is not really trading into it, correct?
This is getting into implementation details, so pay close attention to the parts which seem contradictory. Conceptually anyone who wants sidecoins and has bitcoins or vice versa can simply move value from one chain to the other, in essence destroying bitcoins and printing sidecoins albeit in a reversible process. However, this takes a considerable amount of time to clear -- about 3 days -- and blockchain space which requires fees. So
in practice we fully expect that someone wanting to trade bitcoins for sidecoins will use a distributed marketplace and atomic cross-chain swaps in order to buy one coin with another, by finding a buyer at the other side. Use of the pegging mechanism will be limited to market makers who arbitrage when the price starts to deviate from the peg. Make sense?
Yes, that makes sense.
Why will it take 3 days to clear?
It will be interesting to see (if this is actually implemented) how the distributed marketplaces will act. They will reserve certain amounts of particular sidechains, so how will they know how much? Will their fees for transacting bitcoins for sidecoins and back depend upon the scarcity of the sidecoins they have allotted/reserved? Will such fees be flexible according to the demands for particular sidecoins? We will see, perhaps.
I initially was confused because people were talking about running altcoins as sidechains, and I thought that meant that the same unit limits would be enforced. Now I see that sidechains are simply ways of running your bitcoins as/through specialized protocols that are not available through the primary bitcoin blockchain.
Thank you for your detailed responses.
And I apologize to the Counterparty regulars. It appears we've used this thread to talk about something that is not specifically about internal Counterparty issues, though it might bear on Counterparty.