Are there any serious risks or possible side effects from getting Hepatitis B immunoglobulin? Can anything bad be transmitted through the blood from it?

HBIG. Most serious side effect with the injection of any medicine or biologic product would be anaphylaxis. Rare but possible. Since HBIG is derived from donor serum there could be potential for transmission of infectious disease, but such products are rigorously screened for all reasonable pathogens. Risk of Hep B is far far far greater than either of these.