Yep - they sure do. The DSP will typically implement the “radio” modulation portion of the cell phone. This is why the OMAPs are so popular for this application. The ARM processor runs the GUI OS that the user sees, along with perhaps some audio processing, and the call-stack for phone call setup/tear-down. The DSP takes this and creates the modulation wave-form for the cell call - which is heterodyned up to appropriate frequency.
I take it the DSP is all hardware? I think I once saw a radio built on a software DSP.
I assume the output from the DSP is put through a digital to analog converter before it is fed to the antenna? I don’t really see that mentioned on the chip, but maybe it is done in software.