Whether the energy leaves a system depends on the particulars. If you burn some hydrogen in a bomb calorimeter, the energy will stay inside the calorimeter. This is a "closed system". If you burn it with an open flame, the kinetic energy will warm the atmosphere. This is an "open system". The energy could end up anywhere... you can use hydrogen to launch rockets into space.
Since the energy can end up anywhere and the energy has mass, the mass can end up anywhere, too.
I don't follow how that explains why two identical objects with the same energy content should be expected to have different masses. It seems like this whole thread is trying to argue that chemical reactions destroy mass through examples of open systems that leak energy. A leaky water balloon loses mass, too but that doesn't constitute evidence that atmospheric pressure/tension/etc destroys mass, does it?
If you take a block of steel from the top of a hill to the bottom of the hill and convert all the potential energy to kinetic energy (without loss), why would the mass of the block be altered? It may well be true, but arguments based on applying the mass-energy equivalence is not the answer.
Likewise, if you start with some pool of molecules and rearrange the constituent atoms and bonds, energy must be balanced via kinetics (i.e. translational and rotational energy of and within molecules on both sides of the reaction)--why should anyone expect the overall mass to change from such chemical reactions?
Since the energy can end up anywhere and the energy has mass, the mass can end up anywhere, too.