All equipment is face down.. only you can have your class items... if player a opens the chest with player b's items.. does player a get the equipment? Or does player b flip his gear over reguaedless of place on the map?

The equipment goes to the the appropriate hero class, regardless of map location. I'm pretty sure this is in the on-screen instructions.

It just says to equip it. It doesnt actually say to who that any of the 4 of us playing saw

Right. But it outlines exactly whose equipment it is, and it's pretty explicit in the rules that class equipment can only ever be used by that particular class, and not traded.

It is pretty clear who equips the starting items as they were never lost, just flipped.

Start of quest:
"Each player must flip each of his starting equipment cards facedown. These cards cannot be equipped until you are instructed to do so."

Upon opening chest:
"Rummaging through the assortment of junk and equipment, you locate the missing items belonging to a c lass name . Immediately equip the starting equipment for hero name."

Thanks, Sadgit. You've provided firmer justification.