“block”, “transaction”, “gas” are concepts native to the current paradigm where ordered data and state are tightly coupled. I find it helpful to think in the context of a log shard because that’s the extreme scenario (no state whatsoever), and it forces you to think differently.
This scenario doesn’t immediately apply because there is no native concept of ETH in a log shard (keeping track of account balances is a form of statefulness). One approach to compensating validators for inclusion of logs is via out-of-band payments handled elsewhere (e.g. in the main shard, or in some other stateful shard). I wrote a bit about this in the “Transaction fees” section here:
The collation limit induces a fee market, and fees for inclusion of logs can be paid for out-of-band, e.g. using payment channels from users to validators. (I am exploring an offchain design similar to Raiden where users can pre-purchase generic “stateless gas” to trustlessly compensate any validator for including logs in collations. This is the topic of another post.)
In this model of out-of-band compensation you can also have private knowledge as noted in the original post:
users may have immediate private knowledge of the inclusion of the logs they care about, especially in the context where logs can be included in exchange for out-of-band compensation.