We investigate the internal 3D magnetic structure of dense interstellar filaments within NGC 1333 using polarization data at 850 μm from the B-fields In STar-forming Region Observations survey at the James Clerk Maxwell Telescope. Theoretical models predict that the magnetic field lines in a filament will tend to be dragged radially inward (i.e., pinched) toward the central axis due to the filament's self-gravity. We study the cross-sectional profiles of the total intensity (I) and polarized intensity (PI) of dust emission in four segments of filaments unaffected by local star formation that are expected to retain a pristine magnetic field structure. We find that the filaments' FWHMs in PI are not the same as those in I, with two segments being appreciably narrower in PI (FWHM ratio ≃0.7-0.8) and one segment being wider (FWHM ratio ≃1.3). The filament profiles of the polarization fraction (P) do not show a minimum at the spine of the filament, which is not in line with an anticorrelation between P and I normally seen in molecular clouds and protostellar cores. Dust grain alignment variation with density cannot reproduce the observed P distribution. We demonstrate numerically that the I and PI cross-sectional profiles of filaments in magnetohydrostatic equilibrium will have differing relative widths depending on the viewing angle. The observed variations of FWHM ratios in NGC 1333 are therefore consistent with models of pinched magnetic field structures inside filaments, especially if they are magnetically near-critical or supercritical.