I bypass the compost route in summer and drop them around my rose plants, along with banana peels. The roses love them.Otherwise, yes, all my teabags go into the compost. I never bother to tear open the bags, too much pfaffing.
You can ask at your local starbucks for used coffee grounds and they will usually give you as much as you want....I had to stop getting from ours because I couldnt keep up with the amount they were providing..but if im passing ill pop in have a coffee and leave with a bag full of grounds for the compost it usually has tea bags in it aswell..My compost worms do love an espresso...