Looks like that in order to use two behaviour trees - one for the dialogue system and the other for the rest of AI actions they must share the same blackboard.
That kinda sucks, because I want two characters to share the same behaviour tree for their generic behaviour and each of them must have completely different dialogue.
Both workarounds - copying behaviour trees and creating two ai controllers per character (one for generic behaviour, and another one for dialogues) are not pretty.