Which Attack Succeeds First?

We have mate on both boards.  My partner (right board) has g5+ and Qe6#.  In my board, on Kf5, N@g3 mate or Kh4, g3#.  I was definitely helped by material from the other board, but let us consider how my opponent contributed to his own demise:

Black plays h5.  I've seen this move before, in fact I played it as a teenager.  In theory, if White eventually castles, Black can play h4 and h3 attacking the king position.  If he does not castle, there is the idea of h5 and @g3, threatening gxf+.  In reality, it makes it easier for White to attack.  The Black h-pawn can no longer protect g6 or g5.

I start my attack on the dark squares with Bg5.  f6 is not advisable for Black because it leaves g6 weak:

I am continuing my attack.  I am not only attacking c7, but also d6.

I offer a piece to continue my attack on the dark squares, c7, e7 and f8:

So there was cxd, N@c7+ with the idea of getting the rook for the piece I sacrificed.  He plays Kf8.  I think that Bxc7, Nxc7+ Kd8 is better, making it hard for my knight to get out:

I play @d6.  Black can just play Qd8, saving the queen and only giving back an exchange.  I didn't expect him to give the queen.  Material counts!

Here my opponent tries to start an attack.  This doesn't work unless I ignore it.  After fxe, I am safe.  He would need a knight and two queens to mate me:

Here, I get mated if I take the pawn, but if I don't take the pawn, nothing bad happens:

My opponent plays B@c4+.  This move is used to keep the king from escaping to d3, however, in this case, Black has no attack to escape from.

N@g1+ is fatal if I take on f2, but it does nothing when answered by Nxg1:

After trading on g1, He plays N@g4.  I play Q@f3 to defend and he plays N@g5.  This is getting annoying, as I cannot abandon my defense of f2.  I play Qf5.

My opponent plays @f3+.  He is correct that I can't take the pawn.  If Qxf3, he takes the queen and that mates my partner.  If gxf, Qf2#.  I have to play Kd2: 

He plays B@d6.  He is probably still thinking of Q@f2+.  Maybe he is thinking of a helpmate such as Qxd6, Qf2 B@e2, Qxe3, Ke1 @f2#.  But I play Qxg5 and my e-pawn is still protected.  It does allow Bxc7 which was played. 

I play @e5.  I am hoping to attack with @d6 or @f6:

If I take the knight on d4, he can get me to give up my queen with B@e3.  But what if I take the bishop on c4?  That is what happened.

Black makes a purposeless check on a5 and I interpose.  He plays B@g6, probably to prevent the sacrifice, Qxg7+.  So I play Qxe7+ and my attack begins:

So my attack starts with @d6+ and @e7+.  I have plenty of pieces because I haven't wasted them, and taking the queen has given us the advantage on both boards:

There followed Kf8, R@f8+ Rxf8, exf8Q.  Here he should take the queen.  He goes Kh7.  It is true that the queen would go to the other board as a pawn, but he is leaving me with an extra queen on my board. If  kxf8, @e7+ Ke8, the king is relatively safe and he doesn't lose the rook on h8.

There follows N@g5+ Kh6, Qxh8+ @h7:

Nxf7+ is a clearance sacrifice:

So, there followed Bxf7, @g5+ Kg6, N@f8+.  It's always good to find a knight check from the first rank when the king is out, to help drive it further out.  If Kf5, Q@f4# so Kxg5 happened:

In this position, our opponents resign.  On my board, if Kf5 then N@g3# or Kh4, Q@g3.  My partner is also mating his opponent: