"We have one child - a 9-month-old daughter who is at that yummy stage and is mostly as happy as a larrikin ["A person given to comical or outlandish behavior" --ed.]. She is also at that crying big hot fat tears stage whenever her Dad or I leave the room (even if the other one is still in it). Is this separation anxiety? If so what do I need to know about it and what can we do to help her through it with as little pain and as few tears as possible? We are already playing lots of peek-a-boo with her and talk to her from other rooms so she can hear our voice, etc. I now shower with her in the bathroom with me."
Sounds like textbook separation anxiety to me. And I think you're doing about as much as you can with her. The peek-a-boo is important, because it helps her play with the idea of object permanence. Talking to her and giving her a running commentary so she can hear you when you're in another room is also helping her develop faith that you're still there even when she can't see you. You might also want to talk in the morning about your schedule for the day, just so she knows well in advance what's going to happen next and what you're doing at that moment.
Some people try to get their kids to "toughen up" at this stage by forcing them to separate more than they really can. They think that if they "give in" by keeping their babies close during separation anxiety that the kids will never learn to be independent. That's the wrong way to approach this period, though, because it really is just a stage. It's a normal part of the process of mental and emotional development for kids, and it means that they're starting to understand that they are separate from their parents. Even though it seems like the kids are regressing, once they come through this stage they'll be far more independent than they were before.
The only real cure is time. In another few weeks she'll be happy and will run away from you without a thought. But in the meatime, it sounds like you're doing exactly the right thing to make it as easy on all of you as possible right now.