Actually, both steel and titanium are near perfect spring materials and lose almost ZERO energy from the flexing. So yes, nearly all of the energy does go back into the system.

That said, it IS debatable (and hotly debated) whether that energy returning into the system actually goes back into forward propulsion. For example, one theory is that the frame flex transfers deformation in the tires, which rub against the ground slightly to cause (increased) heat loss.

However, none of the bike manufacturers have released any studies or data that proves (or even suggests) that a stiffer bike is faster.