The Soviet Fighter Units were viewed by the Soviet high command as an extension of the ground Army. Their reason to be, in upper Soviet circles, was as support for ground operations.
The Luftwaffe tried coming in at high altitudes and the Soviets responded by ignoring the high-altitude German aircraft entirely and sending many aircraft at low altitude to attack the German ground troops. The Germans were faced with a simple decision: 1) Continue high-altitude attacks and suffer unsupportable ground casualties, or 2) Come down to low altitude and fight to save their ground troops. Without ground troops, there was no Operation Barbarosa. They chose to save their troops and the Soviet fighters got to dogfight with German fighters at low altitudes.
The P-39s were used as Soviet fighters. Yes, there was some ground attack involved, but the primary ground attack airplanes were Il-2 and Il-10 later on, with a few other Soviet attack types thrown in, such as the Pe-2 (outstanding) and Su-2 and others. The P-39s, P-40s, Spitfires, Hurricanes, P-47s, P-51s, etc. were mainly used to escort the attack planes in support of ground forces, so they primarily dealt with German fighters.
At first, Soviet equipment and pilots weren't very good. After mid-1941, it was hard for a Soviet fighter to live in a German sky over the Soviet Union, and many Germans racked up impressive scores quickly. The Soviets even had to move aircraft production more than 1,000 miles east to escape German bombing. But, by mid-1943, it was getting hard for a German pilot to live in a Soviet-controlled sky over the Soviet Union, Erich Hartmann notwithstanding (he got to the fight in Oct 1942). The La-5/7 were excellent, as were the Yak-3/9 series, and Soviet tactics caught up with the reality of situation. The German attacks in 1942 (Case Blue) and 1943 (Operation Citadel) were failures, helped by Russian weather, and resulted in the German retreat and eventual collapse.
At least, that is my current understanding from books and conversations with former VVS pilots and Russian internet contacts. I suppose it could be wrong ... but I'd need some evidence of it more than a few posts to think otherwise.