Miners should be economically incentive to process the free transactions the same way do process transactions that pays a fee (all transactions in the economic layers must pay a fee). In order to archive that, the miners can redeem the proof of work included in the free transactions. Think of it like the proof-of-work performed by the clients are coupons that the miners can redeem and for native currency that will be included in the block rewards (like a regular fee).
Thus, the interaction with the smart contracts are not free, the gas required to execute the smart contract is paid as proof of work.
How does this coupon system work exactly? Is there a lottery, is there a reward-per-coupon-system? Redeeming the PoW as a classic PoW, with the miners being essentially pool operators and the senders contributing to the pool won't work.
In either case, this will lead to the effect that every block will be full. If nothing else, miners will fill them themselves, just to not miss out on the rewards. This isn't necessarily a bad thing, just something to keep in mind.
Having a separated blockchain for each smart contract has advantages:
The main chain is smaller.
Each blockchain is well scoped
Calculate the global state of smart contract is simpler and faster.z
Busy/popular smart contracts will have longer chains, while the main chain growth at a predictable pace.
The blocks in the main chain will have references to the newest blocks created in the side chains. That means that the side chains are going to be secure through the main chain, simplifying things for consensus (which is happening only in the main chain).
Where is the data of sidechains stored? How are those who store it incentivised to do so?