Study of Ly alpha absorption lines in the spectra of double line-of-sight (DLOS) quasars holds the promise of diagnosing the nature of the structures that give rise to Ly alpha absorption. Based on simulations of DLOS with various separations through a single population of absorbers, four tests have been designed to diagnose absorber mass distributions (smoothly varying with radius or irregular), geometries (spherical or disk/slab-like), and kinematics (isotropic or systematic velocities). Applying the tests to existing data at z similar to 2 we find the following. (1) The observed neutral hydrogen column density (N-H) distributions of Ly alpha lines coincident to both LOSs are consistent with a smooth mass distribution. (2) Observed large anticoincident N-H values are not consistent with a single population of smooth spherical absorbers, which should exhibit sharp cutoffs at small N-H in the N-H distribution. (3) There is marginal evidence that the observed rms value of velocity differences between coincident lines increases with DLOS separation, as is expected for disk/slab-like absorbers that have systematic velocity fields. (4) The observed velocity dispersion along a single LOS is small compared to the rms difference between widely separated LOS, which is not consistent with models of cloudlets moving isotropically within a spherical structure. Overall, only a smooth disk/slab-like model with systematic velocities remains consistent with the inferred properties of a single population of Ly alpha absorbers.