Do you think you have to take yourself first in improv before you can take care of others?
Or do I need to forget myself and by focusing solely on the other person, all will fall into place?
How do I let someone know they are taking too good of care of themselves?
As to the third question, I don't think you can constructively tell anyone they are taking too good care of themselves. It is best to accept it and move on. Work with what you have...John is right, a coach should give the criticism. If you try to tell someone they are taking too good care themselves, they might take it the wrong way and ruin any comfort and rapport you have with them.
For the first two questions: In my limited experience, both!
I will twist the first question a little bit and echo what I've read on this message board. I think you have to bring something into a scene for it to work. So, #1 You have to take care of yourself first. After you've mirrored or endowed a characteristic on stage; #2 Give your complete attention to your scene partner and absorb everything they offer. Which leads to Jill's comment:
I open my eyes wide and pretend to be a puppy, thrilled to see them.
This is the best! I completely agree with this. I hear some improvisers talk about how they are sick of playing with inexperienced improvisers. Also, they only want to play with "good" improvisers or people they know, and I get frustrated. As an experienced improviser you have the power and ability to kill in scenes with inexperienced or "bad" or "new" improvisers.
Alone, you cannot completely carry a scene, however, if you do what Jill does: *open your eyes wide* and LOVE whatever your scene partner says, the scene will be okay. You will be funny and the scene will work. Take whatever comes and embrace the shit out of it.
Will it be the best improv ever? Probably not, but you will have elevated the scene and actively created a beautiful moment with the other person.
Love your scene partners and scenes will work out. When I play with new improvisers it is a lot fun to see them grow and embrace what they give you on stage.
"Great improvisers never look worried onstage. It's not that they became great and stopped worrying, they stopped worrying and then became great." - Miles Stroth