I did this as an experiment last night with Aza. Aza gave me the last pet I needed for my Raiding with Leashes (which was an Anubisath Idol), which was one of his own ones. So I got the achievement! Thank you Az!
So then Az said "Hey, I wonder if..."
"If you cage all the pets for the Raiding with Leashes and trade them to me, I learn them and then pass them back, would that work?"
"Hmm. I don't know but it SHOULD work?"
So we tried it. And it worked. Now Aza has a Mr Bigglesworth, just like me.