If we add a new joint owner to an existing checking account, do we need to provide that new customer a privacy disclosure?

Yes.

(a) Initial notice requirement. You must provide a clear and conspicuous notice that accurately reflects your privacy policies and practices to:

(1) Customer. An individual who becomes your customer, not later than when you establish a customer relationship, except as provided in paragraph (e) of this section;

(e) Exceptions to allow subsequent delivery of notice. (1) You may provide the initial notice required by paragraph (a)(1) of this section within a reasonable time after you establish a customer relationship if:

(i) Establishing the customer relationship is not at the customer's election; or

(ii) Providing notice not later than when you establish a customer relationship would substantially delay the customer's transaction and the customer agrees to receive the notice at a later time.