Does bitcoin mining pool hash randomly and separately

Most mining pools use the stratum protocol. Part of this protocol includes a field for part of the extraNonce, named extraNonce1. The extraNonce is just extra data that is put in the coinbase transaction and used as a nonce in addition to the block’s nonce. Each mining worker can then append their own extraNonce to extraNonce1. This second part is named extraNonce2

Typically what mining pools do is they issue work to each worker with a different extraNonce1. This allows the worker to use the entirety of the nonce field as well as their own space for extraNonce2. The submitted work can also be identified to belong to a particular worker by examining the extraNonce1.

I’m not sure how specific mining pool software determine what extraNonce1 to give out, but an obvious method would be to do so sequentially – each time work is issued, extraNonce1 is incremented so that each worker gets something different to work on.