Humm, maybe i'll have to test your way.
So far, from my experience, only one character needs a bedroll, you double click on it and it heals the entire party. The thing is if a character has their health bar another color than red.... purple for instance, I doesn't heal that character....
Maybe if I go directly on that character and click on a bedroll it will do so, but I have doubts.
Anyway, every character's of mine do have a bedroll...
Characters only share the same bedroll if they are close enough together, and possibly linked too.
Even in single player, I find having 1 bedroll on each character helpful in case they are too spread out after a fight and one of them is near-death.
I have it hotkeyed to "1" for every character so I can get them healed ASAP.
As to the OP's original topic, I would very much like a feature to toggle turn-based mode while out-of-combat. This would make it a lot easier to get my party in position pre-combat, which is often a real chore in single-player mode.