Ordinarily the ducts just sit there passively like water pipes while the glands ooze out saliva to keep our mouths juicy. Squeezing them forces the saliva out in a spray; we can't gleek again until they refill. So to answer your "what's the purpose?" question, there's no purpose.

Is gleeking a talent?

For people who have problems of excessive saliva, spontaneous gleeking while speaking or yawning can be a source of embarrassment. But some people learn to gleek or gleet on purpose. It is a special talent that they can boast of among their friends to gain admiration.

Is Gleek rare?

Gleeking means involuntarily spitting saliva while talking, eating or even yawning. It is caused by excessive saliva excretions by the submandibular gland. And while a whopping 35% of humans can gleek, only 1% can do it on command.