We calculate the renormalized stress tensor of the Hawking radiation due to the massless scalar field in the two-dimensional version of the Oppenheimer-Snyder model of the gravitational collapse with the formula given by Davies, Fulling and Unruh. It is shown that the behavior of the energy density of the radiation seen by a distant observer is not sensitive to the value of the initial radius of the collapsing star even much before it takes the well-known asymptotic form at late retarded times. This transient period lasts for about 20 M in the observer's proper time, where M is the stellar mass. A freely falling observer who approaches the singularity observes the negative divergence of the energy density of the radiation both in the interior and the exterior of the star except on the surface. It is shown that the negative divergence in the interior is closely related to the negative pressure due to the quantum field.