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Poland had many foreign kings and in my opinion it does make sense to include them all into this discussion, even though only a few of them were ruling another country at the time when they accepted Polish crown. The sheer fact that they were foreigners heavily influenced their politics. In one extreme case it even resulted in moving the capital of Poland from Cracow to Warsaw. But it's impossible to generalize - each case should be investigated individually. And that would take way too much time so I will just write about the two kings you mentioned.
Louis rule over Poland was pretty much uneventful. He had several lands to look after so obviously he wasn't present most of the time, delegating power to local lords. In 1374 however he reduced the tax duties of Polish gentry to buy himself some support and to secure Polish throne for one of his daughters. That was detrimental to Poland in the long term* One may draw the conclusion that he was more devoted to the idea of securing the Polish throne than to make royal power strong at the risk of losing that throne for his children.
*But it's not all that simple. One may say that the appeas*m*nt of the gentry through bribery was detrimental to any future ruler who would truly care about Poland and would need the power to make adjustements. On the other hand, it also acted as a safeguard against a ruler who would like to blatantly pursue his own interests at the cost of the Polish agenda. Louis of Hungary didn't have a chance to demonstrate that to any meaningful extent.
Regarding Augustus the Strong of Saxony, he was elected king of Poland through bribery, scheming and civil war. Russia was already a powerful country and influenced the election. Now, that doesn't mean he was poised to be a bad king of Poland :) Well, actually his rule marked a period of absolute disaster for Poland, but still, it's hard to say it was all his fault, or that it was caused by the fact that he was a ruler of Saxony at the same time. The Polish-Saxonian alliance had the potential of being beneficial for Poland since it was naturally aimed against Prussia. I don't think there was any conflict of interest. Augustus even changed his denomination to Catholicism. On the other hand, as a ruler of Saxonia, in cooperation with Russian tsar he started a war with Sweden that ultimately utterly devastated Poland. And he was bribed by Russian tsar to drag Poland into that war. He also used Russian army to deal with internal opposition in Poland. That looks very bad but that was just instrumental use of foreign troops and most probably he would turn against Russia after he consolidated power in Poland. If he had stronger royal power in Poland, his plans might have materialise and his rule of Poland could have been appreciated. Now, I'm writing this from today's perspective when we know that weak central power was the primary factor that contributed to the demise of Polish-Lithuanian Commonwealth. But in Augustus time, lots of Polish gentry hated his guts because of his royal ambitions.
However, recently I came across information that there came one moment when Augustus gave up in his struggle for power against Polish gentry and instead planned to partition the country. If that's true then there is no excuse for that.
Upvote:7
This is a great question! :) Basically speaking, Poland was in state of some kind personal unions for most time from 1370 (death of last Piast king) up to Poland's collapse in 1795.
Louis's rule, as Jake Jay mentioned, was somewhat uneventful, the King resided in Hungary, and reigned Poland through regents (also, worth mentioning, he took Red Ruthenia from Poland and turned it to Kingdom of Hungary).
It's worth mentioning, that all kings from Jagiellonian dynasty, 1385-1570, were de iure elected to Polish throne by the nobility. Kingdom of Poland, and Grand Duchy of Lithuania were at time two separate entities, there were also times they were reigned by different Jagiellonian rulers:
The main residence of Jagiellonian kings was Kraków. This period of Polish history is generally regarded by Polish people very well, it's even sometimes called golden age, for cultural / economical development and mostly successful wars.
After 1569, Poland and Lithuania formed Polish-Lithuanian Commonwealth (Rzeczpospolita Obojga Narodów), which were ruled by common, elected kings. From which, Stefan Batory (Istvan Bathory), duke of Transilvania, is worth mentioning. Reigning from 1576 to 1586, one of most successful Polish kings, praised for victorious wars with Russia over Inflanty (today's Latvia and Estonia).
Sigismund III Vasa, (the guy who moved Polish capital to Warszawa) was for 7 years (1592-1599) king of both Polish Commonwealth and Sweden. Catholic ruler of Lutheran Sweden, he was dethroned in 1599. Despite prolonged wars with Sweden he did not achieve to regain its throne. In opinion of many, his rule of constant wars (also with Russia and Turks) and Counter-Reformation marked the beginning of the fall of Polish statehood. His sons, Władysław IV Vasa (elected in 1632) and John II Casimir Vasa (elected in 1648) kept the title of kings of Sweden.
The rule of Saxons is widely considered to be a time of dissolution of Polish statehood, Jake Jay described it more widely.