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.