Tichu's Dragon is the most powerful singleton in the game. You can count on it to win a trick…though a well-timed bomb can ruin this plan.
As the hand is played out, the singletons are played differently before the Dragon is played and after the Dragon is played. This is particularly true when both members of one team knows where the Dragon is. The possessor of the Dragon can very much take a “wait and see” approach to the trick, after all, they can always step in.
When you take a trick with the Dragon, pass it to the opponent that you believe will go out last; When in doubt pass it to the player who's turn is furthest from yours. At least this way, if your team goes out first, you might be able to reclaim the 25 points from the Dragon.
Keeping the Dragon can help ensure that you have a powerful hand. Though passing the Dragon gives your team an advantage in regards to where the most powerful singleton is. Or it could simply mean “I have weak hand and I want you to call Tichu.” Never pass the Dragon to an opponent.
As with most plays in Tichu, you must weigh the use of the Dragon. If the trick has 25 points on it already (5, 10, K) then does taking the trick justify giving the opponents 50 points? Also, if you have the Dragon and are passed the Dog, what should you assume? Personally, if I receive the Dog and have the Dragon, I will make sure that my partner sees the Dragon early so they can account for that knowledge.