Sidechains do not "mine" using hashpower. Every side chain block is mined by creating a single transaction on the main chain that pays transaction fees to main chain miners in btc like any other transaction.

Basically, if you are a sidechain , you build a sideblock by aggregating side-transactions, charging side-btc. You then turn back and offer btc to main chain miners for them to include a commitment of the block you just built on the main chain.