They can trade them, so make sure you don't give them _all_ your latest techs. But dropping all but the very latest & hottest doesn't hurt IMHO, as it improves relations a lot with your vassals. Neither will they trade every tech they discover back to you.
I find it doesn't matter a lot whether they trade them to other civs, but typically they won't if your attitudes are aligned well. In my last game the attitude and interests for my civ and my vassals very quickly became aligned. Even they Celts, which had nothing but negative modifiers with my civ quickly became incredibly friendly with a few gifted techs, shared religion etc. and hostile towards anyone not under my wings. But make them too powerful and they may decide to fight for their independence when you make your unreasonable demands.
You can ask your vassal to convert to your religion, for instance, to align your interests better. It's a matter of striking a balance between keeping them underfoot and make unreasonable demands, and aligning your interests so that they love you for it. That's the whole fun of it IMO