I have (pre-)settled status and am self employed. Do I need to prove my right to work?

Genuinely self-employed contractors do not need to provide evidence of their right to work in the UK.

The definition of “self employed” comes down to the relationship of employment / contract which is not always straightforward and can be subjective.

Whether the person using your services needs to seek proof of status comes down to the question of whether it is a contract  ‘of’/‘for’ services and that will be based on every unique circumstance.

The following link discusses this distinction and may be useful https://www.gov.uk/hmrc-internal-manuals/vat-taxable-person/vtaxper34000