Imagine I have a bottle with a random quantity of molecular hydrogen, and can apply exactly enough energy to break only one molecule of hydrogen. Is the reaction going to occur or the energy would be shared between other molecules in the bottle?
The questioner asks that exactly enough energy is added to a molecule of hydrogen just to dissociate it and will energy be shared. The answer is both yes and no.
It is better to take a step back and consider what happens when a photon is absorbed. In the case of hydrogen, dissociation must occur from an electronically excited state as it is a homo-nuclear diatomic and does not have a permanent or varying dipole. This means that the ground state cannot absorb a photon's energy from v=0,1,2 etc to any other vibrational level. (I'm assuming absorption into continuum just above v=$\infty$ is zero.) So excitation has to be with a UV photon.
If the photon is just at dissociation, the H atoms will have effectively zero kinetic energy and so not depart from one another. In fact at this point life becomes complicated and interesting. The potential energy between two atoms increases very slowly with increasing separation as one reaches dissociation. In fact it is possible to form Rydberg atoms which may be a micron in diameter when excited to within fractions of wavenumbers of dissociation. This size is huge, about ten thousand times larger than the minimum H$_2$ atom separation, bigger than a protein and approaching the size of a bacterium! Clearly the two atoms hardly influence one another and consequently a small perturbation, a minute magnetic field for example, and certainly a nearby molecule may cause the Rydberg molecule either to break apart or recombine depending on whether some energy is added or subtracted in the interaction.
If there is not enough energy then the H$_2$ will remain in a highly excited vibrational level until it suffers a collision with another molecule and some vibrational energy is transferred to this molecule. Alternatively the electronically excited H$_2$ could fluoresce and any vibrational energy left in the ground state shared on collision with another H$_2$.
If there is far too much energy than needed to dissociate, then the H atoms ballistically fly apart in opposite directions and transfer some of their energy at each collision with other H$_2$ molecules. Possibly reaction also occurs, but H$_3$ is almost certainly unstable.