How wide does a 6x6 concrete post footing needs to be?
Since it is a structure that is not occupied in any way, I believe there are no codes that would govern footings for pergolas.

With that said, in my opinion, any hole worth digging for concrete should be at least 12"X12" or 12"  round. Just to make a point, the posts could simply be set in the ground, using the 6X6 bottom as the bearing surface. Even not knowing your soil type, or bearing capacity. I am sure that would hold anything that may grow upon your pergola, BUT you have the right idea about the concrete. With the concrete poured a little above grade and the post clip added to the bottom of the posts, you have a good set up for long lasting posts.

The biggest thing will be is, you will need to provide bracing so the pergola does not rack.