A simple physical model for long-duration gamma-ray bursts (GRBs) is used to fit the redshift (z) and the jet opening angle distributions measured with earlier GRB missions and with Swift. The effect of different sensitivities for GRB triggering is sufficient to explain the difference in the z distributions of the pre-Swift and Swift samples, with mean redshifts of <z>≅1.5 and <z>≅2.7, respectively. Assuming that the emission properties of GRBs do not change with time, we find that the data can only be fitted if the comoving rate density of GRB sources exhibits positive evolution to z>~3-5. The mean intrinsic beaming factor of GRBs is found to range from ~34 to 42, with the Swift average opening half-angle <θj>~10deg, compared to the pre-Swift average of <θj>~7deg. Within the uniform jet model, the GRB luminosity function is ~L-3.25*, as inferred from our best fit to the opening angle distribution. Because of the unlikely detection of several GRBs with z<~0.25, our analysis indicates that low-redshift GRBs represent a different population of GRBs than those detected at higher redshifts. Neglecting possible metallicity effects on GRB host galaxies, we find that ~1 GRB occurs every 600,000 yr in a local L* spiral galaxy like the Milky Way. The fraction of high-redshift GRBs is estimated at 8%-12% and 2.5%-6% at z>=5 and z>=7, respectively, assuming continued positive evolution of the GRB rate density to high redshifts.