In the recent discussion of Coinbase's sending outage, the question was brought up how the privacy of Coinbase's customers would be affected if Coinbase started to use payment batching instead of sending each payment in a separate transaction.
It seems obvious that any recipient of a Coinbase withdrawal would be able to deduce that the other recipients also got money from Coinbase. However, this information seems to be readily available via publically available chain analysis results already: https://oxt.me/entity/tiid/483238635
What effects on privacy are to be considered when payments are batched? How does this affect the individual recipient's privacy? How does batching affect the privacy of the service provider?