The mm to sub-mm spectrum of Sgr A* at the Galactic center, as well as its polarization characteristics, are consistent with the inner 10 Schwarzschild radii of a tight Keplerian emitting region of hot, magnetized, orbiting gas. This plasma may also be the source (through self-Comptonization) of the X-rays detected by Chandra. It has long been suspected that the circularization region between the quasi-spherical infall at large radii, and this inner zone, is responsible for producing the rest of Sgr A*'s spectrum. In this paper, we report the results of a detailed study of this region, with several important conclusions that will be highly relevant to upcoming coordinated multi-wavelength observations. First, the combination of existing cm and X-ray data preclude the possibility of producing the observed strong 1.36 GHz radio flux via thermal synchrotron within a bounded flow. If Sgr A*'s radio spectrum is produced by accreting gas, it appears that a non-thermal particle distribution is a necessity. This may not be surprising, given that the energy associated with the radial motion is probably dissipated by shocks before the gas circularizes, which can produce the required power-law distribution. Second, if this is the correct picture for how Sgr A*'s spectrum is produced, it appears that the Chandra-detected X-rays may originate either from self-Comptonization in the inner Keplerian region, or from optically-thin nonthermal synchrotron emission in the much larger, circularization zone, extending up to 500 Schwarzschild radii or more. This is a question that should be answered by upcoming broadband observations, since the mm-bump and X-rays are strongly correlated in the former case, whereas the X-rays are strongly correlated to the cm-radio flux in the latter. In addition, X-rays produced in the circularized gas could show periodic or quasi-periodic variations, but not those produced via nonthermal synchrotron emission much farther out.