Bear in mind that most of my maneuvering was done from a defensive position due to the aircraft I flew, and that my experience spanned a period 15-30 years ago. But, I have some perspective that may help answer the question.
First, as a comment mentioned, some are decided at the bar later. (with hands shooting down watches…) We also had a saying: “First person to the chalkboard wins”. This is in reference to the post-flight debrief practice of diagraming each engagement out on the board, and how whoever was doing the debrief may slant their interpretation of the fight to favor them.
The reality though is that despite egos and the competitive nature of practice fighting we never really tallied or recorded "wins" and losses. The objective was always to learn, practice, and improve. There is an honor system in place, and truthfulness is expected to prevail in the debrief. Just like when wrestling or playing other games as a child, you pretty much know when you have been bested. And if you try to BS or make excuses you will be called out and very quickly discredited. So the short answer is that your guess that it is determined by who can sit at their opponent's 6 O’Clock isn’t all that far off.
To get more specific regarding the details of your question, it also depends on the equipment on the aircraft, as well as the level of training being conducted. ACM (air combat maneuvering) can be as informal as a couple of squadron peers with no special gear scheduled for a local 1v1 proficiency flight, (and good naturedly arguing over who won later) or it could be a large scale coordinated exercise like Red Flag, or a Battlegroup at sea.
In the case of the former, BVR or even close in missile shots or guns would be called on the radio. The target aircraft will usually respond “chaff/flares” and continue to maneuver. If the shooter assesses a high probability of countermeasure success, they would respond “continue”. A successful bug-out or assessed simulated kill by the flight lead would result in a “terminate” or “knock-it-off” call, followed by resetting to predetermined positions and altitude blocks for the next engagement. Brief notes and fight diagrams will be drawn up on the pilot’s kneeboard during each reset to facilitate debriefing later. Any disputes can likely be settled by reviewing the HUD video.
If the aircraft is equipped with a CATM (captive air training missile) and/or chaff and flares it will make the training that much more realistic. The CATM will have all the electronics of a live missile, but no propellant or warhead. This will allow the offensive fighter to obtain a missile lock, triggering the defender’s RWR, (radar warning receiver) and give the defender a chance to break the missile lock by dispensing actual countermeasures. (and reinforce use of the correct button – because in a real fight you would want to pump out actual chaff/flares, not key the radio mic!)
In training there are often altitude restrictions on the use of flares, or they may be prohibited entirely depending on the fire danger in the training range area.
For large scale multi-aircraft exercises, the aircraft are usually fitted with an ACMI pod as described here: https://en.wikipedia.org/wiki/Air_combat_maneuvering_instrumentation
Results are live streamed back to the exercise commanders and staff. (Picture Flightaware combined with LiveATC, displayed on a wall sized screen, with a room full of professional pilots and controllers critiquing every move.) These exercise monitors will act as referees in real time, and call out kills as they occur. The “dead” aircraft will acknowledge, change their squawk to a prebriefed code so they can be identified by remaining participants, change altitude to a safe block, and transit to a designated area to hang out (in shame) until the end of the engagement.
The entire exercise is played back later on the big screen in a theater sized debrief room. It can be sped up or slowed down as needed. (As cheesy as the first TopGun movie is, you can see an example of the kind of displays that were in use at the time, and that as I remember were very similar to what was still being used in the early 2000s when I did my last Red Flag)
The debrief lead will provide commentary as it goes, and may pause to ask the pilots for their perspective. For example:
Debriefer – “Hammer15, you were in missile range here when you called the ‘Fox2’, but based on the opening speed of Nail30 we assessed this as a successful bug-out, do you concur?”
Hammer15 – “Yes, I was showing ___ knots, and the CATM broke lock shortly after that…”
