If the intensity is big enough that non linear fibre effects can occur then colliding pulses might result in a lot of spectral scattering. It would depend upon the pulse length as this determines the overlap length and consequently the region that could be driven non linearly. If the pulse amplitudes are the same then for non linearities to occur each must be close to the linear limit.
They will interfere and a standing wave appears, but they will not influence each other at low intensities. At high intensities : see answer of Dymoke-bradshaw.
For the linear regime, you may find a description in: A. Vatarescu, “ Photonic coupling between quadrature states of light in a homogeneous and optically linear dielectric medium”, J. Opt. Soc. Am. B, vol. 31, no. 7, pp. 1741-1745, 2014.
They will interfere and a standing wave forms, but they will not influence each other at low intensities. But at many points in standing wave will lead to a high intensity spots, if these are high enough to kick the non-linear effects in the medium, you will see non-linear influence.