2-way pegging means that value may be transferred at a pegged (fixed) rate in either direction. The "2-way" is in contrast to one-way pegging, which is for example how Counterparty was issued. Unlike proof-of-burn, 2-way pegging involves sending coins to a special form of output which identifies the destination chain and recipient on the other side. At this stage it is similar to proof-of-burn: you have provably made the currency unspendable to you, and in doing so you gain the right to claim an equal number of coins at the destination side chain.
What's new is the return peg: to move those side chain coins back onto bitcoin, you perform the same operation again. First you send the side-chain coins to a special output naming the bitcoin chain and yourself as recipient. On the bitcoin side you take one of those previous burn-like outputs which sent coins into the side chain, and present your proof of having "burnt" sidecoins in order to claim an equal number of real bitcoins.
This special form output is a script which is able to understand an embedded proof-of-spend from another chain, which validates the accounting rules (you need to spend X bitcoins to claim X sidecoins, and vice versa), and which makes sure that claimed coins go to the indicated recipient.
This sortof-is and sortof-isn't fixed exchange rates. If you are coming from an economic background you know the problems of fixed exchange rates, but those problems don't really apply here. The problem with a fixed exchange rate over national currencies is that the two are not in fact equivalent -- you have two issuers that are separately backing each currency. In the case of 2-way pegging however, the two currencies are equivalent. You only ever get sidecoins by making bitcoins inaccessible, and vice versa. It isn't about fixing exchange rates between two currencies, rather it is about using the same exact currency on two different networks: sidecoins are bitcoins, and bitcoins are sidecoins.
thank you for explaining it like this, this is for me totally sound even though you can wonder from a non-technical perspective why you need this special tokens on sidechains, when they are equivalent to bitcoin, but I assume this is like for xcp a technical reason.
Regarding innovation, monetary reward from speculative asset issuance is a failed model. Any innovation that occurs could be cloned on a side chain using bitcoins, freicoins, or some other already issued coin as the native currency. In a free market there is absolutely no reason to prefer the non-pegged alternative.
after that wonderful explanation of 2way pegging, I was assuming a great answer to that part. I have to say I cannot agree less. The reason for innovation (not invention) is rent seeking, if you follow Schumpeter even by definition. This view is after all I know widely accepted over all schools of economics, from the libertarians to the almost ultra-left wing economists, and seen as welfare increasing. To put together a new combination of ideas and capital is probably the most driving force in capitalist developement.
Speculating on this assets is referred to Schumpeter seen as "gods work" because it is allocating money to the best ideas, this money is used for building an entire industry around this idea, creating a slippery slope of innovation, investment and re-innovation. The beauty of the value web following the invention of Satoshi is that it is even self-enforcing the factors (capital, ideas, labour) for creating this slippery slope. This is for me one of the reasons why cryptoeconomics has a huge chance to succeed.
Do not get me wrong, from a non-technical perspective I think it is useful to try to set Bitcoin Protocol as the TCP/IP of internet for money, but you should make sure that there is a sound incentive model for developers, which is encouraging innovation. I think this model is not doing this and probably even leading to the opposite direction. Also reading Adams post on the mailing list for the motivation to do two way pegging was in this way depressing. It sounds, even though I think it is unintentional, like someone who wants to protect the status quo.