Using point-source atom interferometry (PSI), we characterize the sensitivity of a multiaxis gyroscope based on free-space Raman interrogation of a single source of cold atoms in a glass vacuum cell. The instrument simultaneously measures the acceleration in the direction of the Raman-laser beams and the projection of the rotation vector onto the plane perpendicular to that direction. The sensitivities for the magnitude and direction of the rotation-vector measurement are 0.033 degrees/s and 0.27 degrees with an averaging time of 1 s, respectively. The fractional acceleration sensitivity delta g/g is 1.6 x 10(-5)/root Hz. The sensitivity can be increased by increasing the Raman interrogation time, allowing the cold-atom cloud to expand further, correcting the fluctuations in the initial cloud shape, and reducing sources of technical noise. PSI resolves a rotation vector in a plane by measuring a phase gradient. This two-dimensional rotation sensitivity may be specifically important for applications such as tracking the precession of a rotation vector and gyrocompassing.