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We perform a detailed numerical analysis of soliton–soliton collisions within a standard model of the nonlinear dual-core optical fiber. We demonstrated that, almost in all the cases, the collisions are very strongly inelastic, typically leading to formation of a breather that demonstrates violent internal vibrations. In particular, if both solitons are asymmetric and from different branches, the asymmetric breather produced by the collision switches between the two cores with the change of the collision velocity. If the colliding solitons are mutually mirror-symmetric, two asymmetric breathers can be produced. In the case when the two colliding asymmetric solitons are from the same branch, a breather is formed again, symmetric if the solitons are identical. In this case, the breather demonstrates extremely strong internal vibrations, compressing itself, from time to time, into very narrow pulses. We also study in detail an interplay between noise, which is always present in the fiber, and the soliton–soliton collisions. It is demonstrated that the collision always plays a role of a catalyst strongly enhancing deformation of a stable soliton, or destruction of an unstable one, by the noise.