Can bailiffs give my debt information to someone else?

No

Bailiffs work under a contract to the council or authority to recover public debts. They may not sub-contract that work to another company.

If a bailiff has exposed your data to another without authority, then you should report a concern online with the Information Commissioner's Office (ICO) and they will investigate whether a breach has taken place.^[1]

You can sue for material and non material damages.^[2][3]

^{[1] See Reporting a concern to the Information Commissioner's Office
[3] Article 82 of the General Data Protection Regulations
[4] Section 168 of the Data Protection Act 2018}