This article is from the Chihuahua Breed FAQ, by firstname.lastname@example.org (Melinda 'Bob' Casino) with numerous contributions by others.
Opinion seems to vary on this point. Some say it's an old-wives'
tale, while others swear that, although they're allergic to dogs,
Chihuahuas did not bother them.
The best way for someone who has allergies is to spend some time
around a Chihuahua. Some people are allergic to dog hair, while
others are allergic to saliva. Since Chihuahua's are known to lick
the face of their master alot, you'd best be sure you're not allergic
to the latter before investing in this breed.
Ernest Lowell (email@example.com) cautions:
If a Chihuahua happens to "work" for some allergic individuals,
great, but it's more likely that an allergic person will need a
breed with a wiry/curly coat plus no undercoat: poodles, bichons,
and some terriers. Even those breeds don't work for some allergic
people, so it depends on the individual.