Precise measurements of transverse double spin asymmetries A_NN and A_SS in proton-proton elastic scattering at very small values of four-momentum transfer squared, t, have been performed using the Relativistic Heavy Ion Collider (RHIC) polarized proton beams. The measurements were made at the center- of-mass energy √ s = 200 GeV and in the region of t  0.003 ≤ |t| ≤ 0.035 (GeV/c)², which was accessed using Roman Pot devices incorporated into the STAR experimental setup. The measured asymmetries are sensitive to the poorly known hadronic double spin-flip amplitudes. While one of these amplitudes, φ₄, is suppressed as t → 0 due to angular momentum conservation, the second double spin-flip amplitude, φ₂, was found to be negative and small, but significantly different from zero. Combined with previous measurements of the single spin asymmetry A_N, the results presented here provide significant constraints for the theoretical descriptions of the reaction mechanism of proton-proton elastic scattering at collider energies.