You see what everyone else is carrying and doing: the weapon or tool in their hand, their shield going up, a bow being drawn, every swing, and their arrows in flight. An arrow somebody else looses is there to be seen and ducked — it does no damage, and you cannot pick it up.
A health bar appears under a player's name when they are hurt, and lights up for a moment each time it drops, so a blow landed while you were looking elsewhere still registers. Nobody whole wears one — a field of full bars is a heads-up display, while a bar that appears when somebody is bleeding is information.
Players cannot hurt each other. The bar tells you how a
companion is faring against what you are both fighting, which is what it is
for.