Consent to Discuss Clients

Do private practice therapists need consent to discuss clients with others at work?

You don't need additional consent to discuss clients with others at work. Specific levels of consent are assumed in both public and private organizations. This is tied to the roles and responsibilities of the staff. People working within an organization have access to the internal data that is appropriate for their security clearance level. They are allowed to have case discussions with colleagues that have similar roles and responsibilities.
 
Organizations are required to have safeguards in place so that people who don't have security clearance cannot access the information. Leaving a confidential file on top of a desk overnight would be inappropriate because the cleaning staff would be able to read that file. Similarly, having a case discussion in a public cafeteria or elevator would be inappropriate. Security badges, locked doors and passwords are all examples of safeguards that protect the privacy of clients.
 
The key concept to remember is that the data belongs to the organization that has collected it. The organization is responsible for maintaining the confidentiality of the information.