I was one of the people to test this a couple hours ago. Directions to reproduce:
Post doff on the exchange
On buyer character, make sure you have a slot open on the active roster
Search by their name under "Any" (there's no doff section in the exchange yet)
When buying they'll go directly to the active roster. If the active roster is full the purchase will fail (unless they are a doff type that can't be assigned to the active roster, in which case they go to the reserve roster). From there they can be used on the active roster, or removed from it and used on assignments.
I've transferred two doffs this way, a Klingon and a Lethean, and have been using both in assignments as I would any other doff.