Why did the Big Three (Churchill, Roosevelt, and Stalin) distrust each other?
The only reason the US and Britain were allies with the Soviet Union and Josef Stalin was because they shared a common enemy: Germany. They never liked each other, never trusted each other, and even while they fought on the same side, they were conspiring and positioning themselves for a struggle with the Soviets and vice versa before the war was even over. It was an alliance of convenience.
Stalin did not trust either FDR or Churchill, because he thought they were stalling on invading France to open a second front, trying to bleed the Russian Army by not helping as much as they could. Actually, there is some evidence to suggest that he might have been right.
FDR and Churchill were great friends, but Churchill looked at the world like an imperialist, while FDR just didn't think that way. So Churchill thought him naive, while FDR thought Churchill too opportunistic.
A great book about this was just released. It's called Citizens of London.
FDR and Churchill did not trust Stalin mostly because he was a communist. Neither man, but especially Churchill, had much use for communism. FDR did not distrust Stalin nearly as much as Churchill did, though, and that was one reason Churchill was unsure about FDR.
Stalin did not trust either of the other two partly because they were from capitalist countries. Communist ideology says that capitalists are a class enemy who must be fought. Stalin was also personally distrustful of just about everyone.
FDR thought Churchill was too interested in keeping the British Empire intact. Churchill thought FDR did not really understand Britain's strategic needs.
The primary level of mistrust that existed between the leaders was the perception of where loyalty was in the conflict. Stalin had an agreement with Hitler, and there was much evidence to prove that what Stalin was doing to his people was actually far worse than Hitler and Mussolini combined. Yet, FDR and Roosevelt needed Russian support to pressure the Axis forces in an Eastern theatre. The fact that Stalin and Russia were also Communist was also a source of mistrust. This ended up emerging after the War when both America and the Soviet Union resulted in complete mistrust with one another, carving the world between them as democratic and Communist.
I am not sure if the relationship between Churchill, Roosevelt, and Stalin was essentially that of mistrust. After all they were collaborating and cooperating for achieving the common objective of defeating Germany led by Hitler.
Most certainly, in international politics, heads of governments of countries are not expected to absolutely open in disclosing their thinking and long terms objectives. In spite of their common interests, the different countries and their leaders do have other independent objectives, which may conflict with the common agreed objectives.
For example, while Stalin may not openly disclose his resolve to support communism, Roosevelt may not have disclosed his resolve to oppose communism.
However we can say that whatever differences existed between the three were because of the conflicting long term interest of the three countries, and the difference in their political ideologies. But such things are often known to people fairly well, and because of that some distrust to exist between people with conflicting interests and approaches is quite natural.