Study of Lyman-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 Lyman-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 redshifts around two we find that: (1) The observed neutral hydrogen column density (N(HI)) distributions of Lyman-alpha lines coincident to both LOS are consistent with a smooth mass distribution. (2) Observed large anticoincident N(HI) are not consistent with a single population of smooth spherical absorbers, which should exhibit sharp cut--offs at small N(HI) in the N(HI) 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 Lyman-alpha absorbers.