Touch-move...
We cannot watch 400 games.
If there is no independent unbiased witness, and the story is "he touched it /no I didn't", the game must go on, but remind (warn?) the players of the touch-move rules "if you touch a piece you must move it.... if you touch your opponent's piece you must take it..."

Try to keep an eye on the game.

Illegal positions
You name it, it's happened. I was called to a dispute where both kings were in check at the same time. Ask who's move it is, what the last move was, make some sort of decision based on common sense and "trivial analysis". Once again, we respond only to complaints.

Draws
No problems with...

Stalemate
agreement
perpetual check
insufficient material

50-move rule (50 moves by each player without any piece capture or any pawn move by either player) is the most mis-understood rule in chess. Have one player count the moves (count up, not down) out loud.

3-time repetition is impossible to verify unless they are recording. Usually this turns into a draw by agreement. Sometimes you can watch to verify the repetition.

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Parental Interference
Keep parents out of the playing area. We can insist they leave the area; we can insist they leave the building! Even if parents are close, the opponent may feel intimidated.

Interference from other players
Nag, nag, nag

Kids being kids...
Noisy, boisterous, etc. Try to keep them calm. Nag if necessary.