Perhaps the word "peg" is wrong, but I don't know a better word that communicates the idea. Simply "equivalent"? It is not a "peg" like with fiat currencies where one currency is manipulated to always match another. It is the same currency. It is like if you have 1 USD as a dollar bill, and you deposit it into your bank account. Then you no longer have a dollar bill, but you have 1 USD in your bank account. This can be changed back into a dollar bill by going to your bank and withdrawing it. You can trade back and forth infinitely PaperUSD <==> BankUSD, and the bank is never going to try to give you more or less than 1 BankUSD for 1 PaperUSD.
I'm looking at side-chains as encapsulating cocoons or pods rather than pegs, since it seems that the value contained in each does have the potential of varying.
1 BTC -> PodA (1.0)
PodA has been instantiated with 1 BTC and is valued as such (this is similar to Proof-of-Burn so long as PodA exists). Now assume that demand for PodA increases by 50% which would value PodA at 1.5 BTC. This would mean that an additional 0.5 BTC is transferred to the PodA side-chain.
0.5 BTC -> PodA (1.5)
It's dependent upon simple demand (inflow) and supply (outflow). What's tricky is that this is all instantiated within a static parent supply (Bitcoin's 21mm divisible units).
The key to this working is the atomicity maaku mentions, otherwise the transactional barrier is no different from the Btcpay method in Counterparty.
1.0 BTC in PodA will always be valued at 1.0 BTC in mainchain. Why would you pay 1.5BTC for something that you can get for 1.0BTC?
You seem to be taking a "market cap" view with the Pod discussion. The difference is in the assets, and the asset issuance model. Assuming that assets may be issued within the sidechain, the "value" of any asset within the sidechain (and thus the sidechain as an aggregate) need not be related to the size of the BTC-currency used to create it.
Somebody could create assets in a sidechain that represent hundreds of millions of dollars worth of real estate, primed with 1BTC. The lots of land are transacted by a colored coins method: each lot is denoted by 1 Satoshi (but it is obviously valued much higher). Any BTC coming into this sidechain would presumably solely be used for the purchase of such lots.
So take RealEstatePod primed with 1 BTC, each lot of land = 1 Satoshi. Total BTC in this Pod is 1 BTC. (But "value" is many millions of $$$.) Now Buyer wants to purchase a lot, and transfers 100 BTC from mainchain into RealEstatePod for purchase of 1 lot. 100 BTC changes hands from buyer to seller. 1 Satoshi changes hands from seller to buyer to denote ownership of land. Pod now contains 101 BTC. Seller withdraws 100 BTC to mainchain to cash out. Pod now contains 1 BTC.
The value of the assets denoted within the RealEstatePod did not go up or down when the 100 BTC were transferred in and out. The value of 1.0 BTC stays the same across both chains. It is only the specially denoted Satoshis (or whatever other increment the asset issuer chooses) that hold a different value.
Theoretically, the dollar-value of any one sidechain could exceed the dollar-value of all bitcoins in existence. There's no theoretical problem with this.