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: 
Here is a lower level explanation: 

Two photon physics , i.e. photon photon interactions exist and even gamma gamma colliders are proposed for experiments. Here is a two photon Feynman diagram, lowest order:

[![photon photon](https://i.stack.imgur.com/ttmYqs.png)](https://i.stack.imgur.com/ttmYqs.png)

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> [A Feynman diagram](https://en.wikipedia.org/wiki/Two-photon_physics) (box diagram) for photon–photon scattering, one photon scatters from the transient vacuum charge fluctuations of the other
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With [electromagnetic couplings](http://hyperphysics.phy-astr.gsu.edu/hbase/forces/funfor.html) at the vertices, this diagram when calculated gives a very low probability due to the 1/137 value of the coupling constant Only at high energies due to the functional form under the integrals can a measurable probability be predicted.

Suppose that one has quantized gravity , and the hypothetical graviton exists. The corresponding coupling is 6\*10^-39 . So the equivalent gravitational interaction diagram would be about 10^35 times weaker than the corresponding electromagnetic one, just at each vertex. Thus one is talking of infinitessimally small gravitational effects, not attainable in our laboratories. In cosmological models such energies can be hypothesized and in the primordial soup of the [Big Bang](https://en.wikipedia.org/wiki/Big_Bang) it will be part of the gravitational interactions that bounds the universe but these are theoretical propositions.
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.