Agree with Pasty
Both players are on the scorecard A & B, both gross scores can be recorded under their respective column A or B - best stableford points, dependant on handicap allowed (3/4 of full), is then put in points column.
Only the best points score need be put down - in the correct column for the player, so the handicap allowance can be checked - but some guys write both and there is no problem with that.
On a hole where both hole out for the same points it is generally accepted that the score recorded is the first down.
. . . and it is wasting time if your partner has already holed a par and you want to also hole out - just a matter of etiquette though - but will p!ss of your fellow competitors enough to get them to make a song and dance about whose score is recorded.
Paul, you should have taken the hint