No, hivemind doesn't write to the blockchain itself. It can only write "2nd layer" information, derived from 1st layer (blockchain) data. When you follow someone on hive, your hive client app actually generates and signs a custom_json transaction (with a follow command embedded inside) that is sent to a hived node to get incorporated into the blockchain itself.

Hived just adds this new transaction to the blockchain, but it doesn't do any analysis/interpretation of the follow (because custom_json operations don't "do" anything as far as the first layer is concerned).

But hivemind sees this operation in the blockchain stream it gets from a hived node and sets the appropriate 2nd layer data so that API calls to hivemind return the appropriate results based on your follow list.