Requesting pushback most likely depends on the situation that the aircraft(s) are in.
For a simple airport - say, a class C, or D airport like BNA or PHF respectively, you would just have to call ground on the appropriate frequency and request for a pushback, since the taxiways are past the active movement area boundary.
For a more complex airport, mostly class B, there would be seperate ground and ramp control frequncies.
at ATL, for example, each concourse has a "tower" that gives them their clearances, and synchronizes the pushbacks with the neighboring concourse, so as to avoid a conflict (which is a not a good thing at a class B, since it is best to keep moving). in PHX, HP has a ramp tower all their own to synchronize their arrivals, departures and the such. This very same control does the clearances, allows for pushback, and assigns the gates (as well as other things inside the company itself). I am unsure of the other terminals and/or concourses, since i have never been to those before
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