Excellent question! In the US Army Air Forces, after flying a mission in which they claimed to have shot down enemy aircraft, a pilot would be debriefed with the rest of their squadron. After the debriefing, they would file an Encounter Report in which they detailed the circumstances of the shoot-down and gave specific evidence. An Encounter Report also usually required a different pilot to give supporting evidence to confirm the pilot's claim. After submitting the report, the pilot's report and gun camera film was reviewed, and the kill(s) were either awarded as destroyed, probably destroyed, or damaged. In the case of a 1v1 dogfight like you mention, the supporting statement was nullified, so the only thing to go off of was the pilot's personal account and the plane's gun camera film. A particularly hair-raising example in my last source below.

A typical Encounter Report read something like this:

CONFIDENTIAL

SIXTY THIRD FIGHTER SQUADRON

AAF STATION F-150

A.P.O. 558 U.S. ARMY

Personal Combat Report

VIII Air Force F.C. No. 1443A

2nd Lt. Samuel K. Batson

a. Combat

b. 23 December 1944

c. 63rd Fighter Squadron, 56th Fighter Group

d. 1145-1215

e. NW of Koblenz [Germany]

f. CAVU [ceiling and visibility unlimited]

g. Fw 190

h. One Fw 190 destroyed

i.

I was flying Daily Red 2 [Red was the callsign of the 63rd FS; In late 1943 fighter groups expanded in size from 75 to 108 aircraft and each squadron was divided into 2 groups; Daily was the call sign of the 63rd's A Group; Lt. Batson was flying the 2nd plane in a flight of 4] on Lt. Trumble's wing with Lt. Daniel and Lt. Hoffman as number 3 and 4 men. We were at 27,000 feet NW of Koblenz when the group made contact with e/a [enemy aircraft] and we were all ordered to drop our tanks. My leader then started orbiting to the left and we continued turning down to 18,000 feet, where there were many P-47s and Fw 190s making it very hard to distinguish friendly from enemy. We started an attack on an Fw 190 and he broke in my direction. I turned sharply to the left and was able to get behind him at a fairly long range. As I started to close he made a sharp climbing turn to the right, where at this point, I fixed him in my K-14 sight and began firing, the range being approximately 800 yards. He continued his climbing turn and I kept firing but i did not observe any strikes until he had almost completed his turn, and I had closed to approximately 400 yards. I then saw strikes all over his airplane and his engine caught on fire. I then broke off the attack and watched him go in a gentle spiral earthward, completely on fire by this time. I watched him go all the way down, hit the ground and explode, setting the surrounding woods on fire. I did not see the pilot bale [sic] out.

I claim this Fw 190 destroyed.

j. A/C No. 44-19960 [http://i.imgur.com/Mkz9K6K.jpg]

Ammo fired - 891 rds. A.P.I. [armor piercing incendiary]

SAMUEL K. BATSON

2nd Lt., Air Corps

SUPPORTING STATEMENT

I was leading Daily Red flight with Lt. Batson on my wing. I was firing at an Fw 190 when another 190 came down at us from 9 o'clock and high. He overshot our flight and Lt. Batson opened fire on him. I saw many strikes on the e/a and it burst into flames.

PERSHING B. TRUMBLE

1st Lt., Air Corps

Sources:

56th Fighter Group

http://littlefriends.co.uk/56thfg.php

56th Fighter Group combat reports from the archives of M. Williams

http://www.56thfightergroup.co.uk/combatreports/adrianse-carter/index.html

354th Fighter Group

http://www.354thpmfg.com/dedications_groupaces.html

404th Fighter Group, and an interesting account of a 1v4 dogfight

http://www.winkton.net/pages/mccolp~1.htm

Many aircraft had gun cameras installed automatically taking picture or video when the trigger was pressed. This would be used as proof of a kill.