The kind of door that you describe already exists. They are commonly used, for example, in restaurants between the kitchen and the dining area. Servers, who have the hands full, merely need to push their way through. To ensure that two people going in opposite directions don't crash into each other, they are always double doors, and the users adopt a convention (e.g. "always use the door on the right").
However, the problem is that doors that swing both ways offer poorer security. They can only be locked or held shut by via a protrusion into the top of the door jamb and the floor. Either the user would have to reach up to the top of the door to latch it, or there would have to be a complex mechanism to extend the latching mechanism down to customary height. Even then, the lock wouldn't be nearly as strong as that of a door that rests against the entire door frame on three sides.
Oh, and the whole thing would be more vulnerable to bearsbears.
