Personally, I'm pretty conservative. On one side, I've played a priest and a resto shaman, and I will throw a lifetapped warlock a heal only when I'm at full mana and impatient to start the fight. Otherwise, he should be eating-- I shouldn't have a responsibility to heal him after he's hurt himself. Now, some guildie warlocks I like, and I'll do it anyway. But I don't think there should be a responsibility there. As for pets, I'll heal them in a fight 1) when we need them, and 2) when I've got mana left. If I'm healing the main tank with little to spare, sorry, but your kitty is gonna die. But if your VW is the only thing between me and a melee add, I'll keep him alive. Still, I don't think any pet class should ever count on heals unless the situation has called for it. Pet classes have ways to heal their pets for a reason.
On the other side of things, as a warrior, if I'm out of a fight and still hurt, I make it a point to be eating or bandaging before the priest even has a chance to look at me. With mages around and food extremely cheap, there's no reason a priest should have to waste mana in between pulls, even if it's as simple as throwing a HoT on. Carry some bandages (no excuse for not having runecloth), or eat that bread ASAP.
Of course, there are lots and lots of exceptions. Most priests I know will heal me anyway out of combat. And as a priest I'll usually throw off a few extra heals before I drink, including a few to the pets. But as for responsibilities, I think there should be as few as possible on both sides. Your thoughts?