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Based on a German question-answer site:
The left handed based falconry is based on the falconry on horses. You have the falcon on the left hand and the right hand is free to hold the rein. In meantime there are people with the falcon on the right hand.
Other answers: You need the right hand to hold a line. Or they need a free hand and most people are right hander.
There is the similar explanatory statement on another website. (page 5)
Upvote:0
All falconers in Central Asia, not just Mongolia, carry the falcon on the right arm (the "good" arm), I would guess because that is the stronger arm. You can read, for example, the account of William of Rubruck (c. 1220 β c. 1293) who wrote definitively that the falcon is always carried on the right arm and it is the same way throughout Persia as well.