On a joint checking account, can either owner add an authorized signer to the account or do both have to agree to it?

Whether it's "required" that both parties agree is purely a function of the law of your state. Is it a "best practice?" Absolutely.

An agent, convenience signer, or authorized signer is, in it's most practical interpretation, a limited power of attorney respecting a single account at a single institution. As any power of attorney ceases on death of the principal, if only one signatory appointed the agent then the agent's power ends with the death of the person who made the appointment.

I've never seen and do not expect to see a bank that has the capability of realizing that only "Dad" appointed "Local Son" on his joint account with Mom and, when Dad dies, informs Local Son his signature is no longer authorized. On the other hand, if Local Son was appointed by both parties his appointment remains in effect until the last one dies.