Question: Say two photons moving beside each other through space at speed of light, since it carries energy it can bend spacetime. So both photons will exert gravitational force on each other at speed of light however so slightly, say strings theory is right that gravition really do exist. I'm thinking if the force carrier of gravity moving at speed could exchange between the two photons? Of course I'm aware this whole stuff makes little sense so what's am I missing here?

Answer: 
Photons don't move at all. What changes is the probability to find an excited photon state at a spacetime point. If gravity were just another field, then the existence of a photon state in one spacetime point would create excited graviton states around it. This is the usual formulation of quantum field theory on a preordained background. 

Unfortunately gravity doesn't quantize this way. Gravity is the distortion of the background metric itself, rather than an excitation of a field that lives on the background metric. 

I do not believe that string theory has achieved the required manifest background independence, yet, but I am willing to let a string theorist correct me on this. In absence of this string theory can still use a perturbative approach to the problem that one can hope gives the correct results without requiring a background independent mathematical formulation. 

Having said that, the coupling between single photons and gravitons is incredibly weak until we get to extremely high energies. If the Planck scale exists (a big if), then we will have basically no way of producing photon energies high enough to see the production of high energy gravitons in our accelerator facilities anytime soon. At this point the question if gravitons exist is therefor experimentally not tractable, leaving little else but guesswork for theoreticians.
Because in my view we have a coupled photon-graviton.

The Photon – Graviton pair (coupled) has the same speed and frequency, and the photon energy divided by the graviton energy is the electromagnetic energy divided by the gravitational energy, the electromagnetic force divided by the gravitational force.