No.
Person to person gun transfers across state lines are not legal, even if they physically transport the firearm themselves (either of you). It does not matter if it is a long gun or a handgun. Without a FFL handling the transfer, firearms can not change ownership between two people who live in different states.
If it is a long gun, the transfer must go through a FFL in either state.
If it is a handgun, the transfer must go through a FFL in Missouri (the home state of the buyer).
