We create streaming ultracold neutral plasmas by tailoring the photoionizing laser beam that creates the plasma. By varying the electron temperature, we control the relative velocity of the streaming populations, and, in conjunction with variation of the plasma density, this controls the ion collisionality of the colliding streams. Laser-induced fluorescence is used to map the spatially resolved density and velocity distribution function for the ions. We identify the lack of local thermal equilibrium and distinct populations of interpenetrating, counter-streaming ions as signatures of kinetic behavior. Experimental data are compared with results from a one-dimensional, two-fluid numerical simulation.