The AI does give its techs to vassals. Always as a gift, never as a trade. It appears to be limited to one gift per turn so it may take a while for all techs to be gifted to a vassal. The consequence is that a vassal can thus outresearch its master (vassal gets all techs gifted from master but does not gift techs to master).
I do give techs to my vassals but my generosity very much depends upon who my vassal is and its capacity for research.
If I have an aggressive vassal, such as Montezuma or Shaka, I gift the bare minimum they need to defend their territory. BTW I only accept the aggressive civs as vassals by way of utterly defeating them in war and only when they are too small to have any chance of shaking of vassalage. Even then I seldom vassalise them as I would rather completely destroy them. (Their existence antagonises the peaceful civs with whom I like to maintain good relations and they have a very high desire to break the vassalage).
The peaceful civs, such as Elizabeth and Hatshepsut, are always welcome as vassals. I find them more reliable vassals. If they are small I am generous with tech but I may withhold space techs to avoid the possibility of them beating me in a space race. I also keep in mind that they will trade any tech they research so I might tell them what to research. I may hold back a tech which my enemies have and tell my vassals to research that tech.
Depending on the situation, I might use a vassal as a tech trading partner. I research one tech while they research another and then the vassal gouges me for a trade. I have tech brokering turned off so I research the tech I least want my vassal to tech whore to my enemies.
What I do depends thus upon the situation.