The time period I'm interested in is 17th/18th century Europe. The Dukes of Prussia/Brandenburg at one point would have answered to the King of Poland and the Holy Roman Emperor as Prussia was in Poland and Brandenburg was in the HRE. What would happen if those two nations went to war?
The Hanoverian Kings of the UK were also Electors in the HRE, would the Kings of England be subordinate to the Emperor? How would the different authorities/sovereigns administrate the lands? The Seven Years War saw the UK and Austria at odds, how did that work?
Wait, as far as I know Hannover was never an elector, no?
It has been a while since I read it, but in Christopher Clark's Iron Kingdom he suggests that the original reason that Frederick I of Prussia sought the title 'King in Prussia' (not of - in) was so that he could better exert authority over subjects who defaulted to Polish (I think it was Polish? Could have been some other dynasty) laws/rulers in certain disputes, limiting the elector's influence.
So perhaps the answer to your question is that for the subject with 'split' loyalty, they try to find a deal that best suits them (e.g. if the elector of Brandenburg is giving you a hard time lean towards the other side, then back again as necessary). For their nominal superiors competing for their loyalty, it was a matter of trying to find ways to coax them along or exert your authority - be that by taking on a more powerful title, or more direct means.