I work for AT&T as a network design engineer on the IP side of things (mobility side).
As was mentioned it depends - to be honest aside from some work in AT&T labs most of what the telecoms do (e.g AT&T, Verizon, Sprint, etc) is not electronics design in any way shape or form. We don't design/deploy our own stuff, we buy it from companies like Alcatel-Lucent, Ericsson, Cisco, Tektronix, etc.
If you want to do electronics design in telecom you'd be better off looking at telecom vendors like those I listed rather than the telecoms themselves. In the service provider space you're going to be looking at network engineering, RAN engineering, IT, DBAs, etc.