PHP - Is it possible to write a conditional that runs when a different function is executed by a different user?

I have an existing function (#1) on my page that runs when a form is submitted. I want to run a new function (#2) whenever #1 is submitted. I've looked at AJAX to work with the form, but that won't work given my current environment. I was previously just running #2 on a loop every 1 second, but you can imagine how terribly inefficient that is.

For clarity: I am not asking how to accomplish a form submission that isset can handle. I want to run a function on one user when another user submits the form.