blink1 | Official software for blink USB RGB LED by ThingM

As @Simplicius said in most cases for what you're asking for you need a server. I think you may be looking at the problem in the wrong way. What is your ultimate goal? Is it a 2 player game that multiple people can interact with? If so doing it with a single page is not possible without something hacky like localStorage or sessionStorage. Somehow I dont think that's what your teacher has in mind though. I would revisit the problem and really understand what's being asked of you before moving forward.

In blink2() you test the global variable i, but you never change it.

Note aside: blink1() manages LED2, while blink2() manages LED. Perhaps a more consistent naming would help, and the same applies to the variables i, x, y and count.

There is no restriction to use more than one interrupt concurrently, but when of them executes, the others are temporarily blocked. It is not your problem here, your code is fine; but if you want an interrupt be able to be interrupted in turn, for low latency, you must re-enable interrupts in the (relatively) "slow" handler.

Last suggestion: where you have "if (y==100)" and lately "y=0", if you move "y=0" just below "if (y==100)" readability improves.

All the rest seems ok, but I would double-check the setup of the timers; I don't have at hand the datasheet, may be there is a flag to clear in the interrupt handler (I can't remember). Given that you say that a single timer works, I suppose you know what to do, two timers should run without any problem.