I still can't quite fully wrap my head around how personal unions worked in the medieval-early modern period. Entities such as the Crown of Spain controlling independent Castille and Aragon, or Denmark-Norway, or Charles V as King of Spain/Duke of Burgundy/Holy Roman Emperor. How much independence did the seperate nations actually have? Would these seperate nations ever be distinct enough to declare war or embargo each other against the wishes of their shared monarch?
While Charles V was Holy Roman Emperor he had very little input on the politics within it. Charles realised that it would be impossible to rule both Spain and its Empire as well as the Habsburg lands in eastern Europe. So his brother Ferdinand was given control of these territories while Charles ruled directly in Spain. Charles V kept the imperial title for the prestige associated with it, as well to not be outranked by his own younger brother.
Two states in personal union would never go to war, but quite often one would be unwilling to help the other, with Charles often offering fairly lacklustre support to his brother against the Ottomans in the East.
Another example is Britain and Hanover between 1714 - 1815, with Parliament often refusing to send money and troops to protect Hanover during wartime, arguing that what was happening in Hanover was the King's personal business as Elector of Hanover, and was of no concern to the people of Britain.
Even when Spain was united under one Crown there still remained the need to seek money and supplies from the Cortes of the different kingdoms. It was very much a case of treating the sovereign as of he ruled only one Kingdom, and ignoring his other domains. When asking for money in Barcelona Charles V or Carlos I as he would have been called, did so in his capacity as Count of Barcelona, not as Holy Roman Emperor. This is one of the reasons why he often gave poor support to his brother against the Ottomans, with his Spanish domains already funding his seemingly never ending wars with France, they weren't willing to send money and troops to the Balkans to fight the Ottomans as well.