How did Churchill and Roosevelt compromise their beliefs with Stalin at the Yalta conference ?
There are certain implications about the way that Roosevelt and Churchill may have compromised their "beliefs" in allowing Stalin to have his way in Eastern Europe, but in many ways they shared a great deal of common ground about how they felt about certain ethnic groups (remember both Churchill and Roosevelt imprisoned German refugees, many of the Jewish as well as Japanese, etc., though thankfully neither of them ordered millions of them to be killed!) but it wasn't a matter of two great idealists giving in to a communist monster.
If you read their correspondence prior to the war (an interesting and certainly strongly biased yet valuable account of some of their writings can be found in Nicholson Baker's book "Human Smoke") they were in many ways very interested in getting involved in the war and obviously felt that Stalin's participation was vital (and if you look at the way the Eastern Front destroyed Germany's armies you have to understand how vital it was) to their desires to win the war. So if their belief was that they needed to win the war, they didn't really compromise a great deal in allowing Stalin free reign in Eastern Europe.
Roosevelt was ill at the conference, and definitely not his full formidable diplomatic self at Yalta. But he and Churchill knew full well what Stalin was and what he was capable of. They knew he was not to be trusted. It was on this front that they compromised. They (mostly Churchill) gave Stalin a free hand to take both Eastern Europe and Berlin during the war, with the influence to set up postwar governments. The agreement they trusted him on was that in ten years, there would be free and fair elections in those countries. They knew better, or should have, as Stalin simply rigged the elections and installed puppet communist regimes. As Churchill would later famously remark, "An iron curtain has descended across Europe" separating the communist East from the non-communist West.
Both Churchill and FDR believed very strongly in democracy. Of course, Joseph Stalin did not. Stalin wanted to dominate the countries of Eastern Europe, no matter what the people of those countries wanted. This, of course, goes against the idea of democracy. So you would think Churchill and FDR would have opposed the idea.
But instead, they essentially told Stalin he could do what he wanted in Eastern Europe. They did this because they felt it was more important to have his continued support for their goals in ending the war against both Germany and Japan.