Your asking "Does anybody have an example of a functional relationship where both people are seeing a psychiatrist or therapist?" is a bit unfair.

Earlier, therapists were meant for the kooky people who heard voices or went in the complete opposite direction of the norm.

Today, more and more people have started taking therapy for any minor dysfunction in their life. And that, I believe is a very healthy sign. Seeing therapist separately is helpful if you are dealing problem as an individual. Say, you saw something traumatic as a child - like seeing someone die....it's something you probably want to deal individually, and not as a couple.

Psychologists believe we ALL have secrets. Different people have different level of secrecy. Maybe you share. Some people may like to share less.

Secrets which doesn't affect you being in a couple should be fine.