If radioactivity can cause cancer, do cancer radiation treatments radiate the people around the patients?

No, except. External beam radiation does not make the patient radioactive and they do not emit radiation. The exception being the few cases in which radioactive material is implanted in the patient, e.g., radioactive seeds for treatment of prostate cancer.
NO. Patients who received radiation, the radiation energy like gamma rays dissipates in the tissues immediately, and patients can not re emit radiation, and safe to other people. Similarly implantable so called ' radiation seeds' are safe to other people and environment.