VS CPU players it would indeed make a good training session that way ~
The possible problem I see in doing it the teleportation way is this:
If it's like there's only 2 players facing each other and the winner is teleported to another room where another player THEN spawns, most players would have to wait a long time, furthermore you'd (possibly) have to make the player regain health and lose his golden gun (because it would be unfair otherwhise).
If it's like whenever a player wins a duel he's teleported then you'll have the following problem: a player finishing his duel way faster would still have to lose his golden gun but also he would have more time to find the golden gun in the new arena and could already be holding it when his enemy is teleported in there -> the first player would have to be frozen in place until his enemy shows up.
So if you do it the second way, you could go for roundbased action.
Although I'd like to mention that maybe some achievements should be disabled for this game mode, since it's easier to obtain them, because it's always 1 on 1 with full health.
EDIT:
Limited ammo should keep players from spraying, let's say max 3 or 4 clips of ammunition in total.
For a possible training mode (when AI is implemented in future) you could use Trent Easton from Perfect Dark as Lvl 5 bot model (or as an unlockable superbot combined with an achievement if you beat him, maybe even give the player the ability to select him in multiplayer when you beat him)