Perhaps the itchy bird of the pair cannot develop the antibody to rid the body of the bug. Treatment should never be attempted without proper veterinary diagnostics. Fecal direct smears with Lugols iodine sometimes shows the organism. A plethora of other tests are described by avian veterinarians, none of which produce reliable results. A complete blood count and Gram stain of mouth and feces should be performed to help rule out other diseases.
Some birds do very well with one treatment series.
Others relapse periodically and need treated again. Environments should be cleaned well, and never should a bird be allowed to pick over feces in the bottom of the cage or flight. Giardia is also a disease of people. To our knowledge, no reports of infection from man to bird have occurred. In my experience, lutino cockatiels are more prone to Giardia than others.