When an antiserum is injected to a person to protect oneself from a certain disease, the antibodies in the antiserum come from another organism. The question is:
Why don't the injected antibodies cause an immune response in the individual?
That's a good question - because its certainly possible to get antisera against antibodies.
Sometimes antisera is actually human antisera, so there would be no rejection at all. But that is for extreme cases, e.g. serum transfusions from convalescent Ebola survivors to current Ebola patients.
One would expect that you could only give an antiserum once and then it would be less effective each time you get it. Since antisera to snakebites is the most common application, where its quite possible to be exposed to multiple bites over one's lifetime.
I found one reference to a horse antiserum treatment that's being used. It makes reference to a source of antiserum which sounds like its been designed to reduce the antibody antibody immune reaction:
Even before the availability of these new treatments, its probable that most patients can endure multiple anti serum treatments from the same source animal. Even goat, horse and rabbit antibodies are pretty similar to ours and the reaction will not be as strong, in addition to which, the immune system will tend to clear out the antigen-xenoantibody complexes from the blood.
I can't find references, but my guess is there must have been some cases of rejection and reaction otherwise they would not have bothered to improve the serum treatment.
Update: just read this little blurb in scientific american, that describes how venom antisera will occasionally cause deadly reactions, while most of the time simply disarming the venom. The new generations of antivenoms described have the Fc portion of the antibodies removed. This eliminates the adverse immune reaction but I also think also eliminates any immune resistance the patient might get out of the antisera. The Fc portion of the antibody is the part which our immune cells bind to. In this case the new antivenoms aren't anti sera exactly, but they do use the antibodies to deactivate the venom toxins and crosslink them to make them less effective.