More likely than not the Y-cable is not for splitting a 6.3mm stereo signal to two 6.3mm stereo outputs.
At any rate, it sounds like you want to hear one amp left, another right. When using the normal connectors of your headphone, you'll be making a ground connection with your amplifiers and more likely than not they have independent grounds already. That gives you a ground loop which is cause for hum.
Now if your headphones are isolated from your head, it's possible that they don't have a common ground except for the connector. If that's the case, it might be an option to cut off the connector and solder two independent mono connectors to the two separate shielded cables at the amp end. This only makes sense when the cables are of the kind only joined by plastic and could be pulled apart without having any wires show.
With two isolated and separate earphones, you should be able to connect to two different amps without problems. However, the amp output will tend to be for a stereo connector (TRS, tip ring shield) with tip and ring both connected to the output. If you insert a mono plug, it will shorten ring and shield, consequently shorting the whole output.
More likely than not that's what you currently are doing with your Y cable.
So whatever you do, you'll need a full stereo TRS plug in the headphone socket of your amp that does not connect R and S.