Is giardiasis being transmitted through the mouth?

Yes: The infective stage of this paracite is a cyst that matures after reproduction in the intestineof the symptomatic or asymptomatic host. The host may be human or animal ( wild or domestic). That material ends up in the water and may pass unseen into the mouth from the contaminated hands or thru ingestion of tainted water. Day care centers/wadeing pools are common risk factors.

Partly: It is a fecal-oral transmission. Hand washing is critical to prevent transmission.